ENUMERATION OF STANDARD PUZZLES

GUO-NIU HAN

I.R.M.A., UMR 7501 Université de Strasbourg and CNRS 7 rue René-Descartes, 67084 Strasbourg, France

                        Email: guoniu.han@unistra.fr

Abstract. We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which the enumeration problems can be solved by explicit formulas or by classical numbers, such as binomial coefficients, Fibonacci numbers, tangent numbers, Catalan numbers, …

1. Introduction

We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules, and study their enumeration problems. The topic of the paper may be classified as belonging to Enumerative Combinatorics, since several of those standard puzzles can be solved by explicit formulas or by using classical numbers, such as binomial coefficients, Fibonacci numbers, tangent numbers, Catalan numbers, …

The general definition of a standard puzzle is inspired by the following classical topics in Enumerative Combinatorics: (1) polyominoes [11] ; (2) standard Young tableaux [14] ; (3) permutation patterns [12] ; (4) doubloons, which were introduced recently [5, 2, 3].

We do not pretend to establish a general principle that will make the enumeration of all those standard puzzles possible. We will only provide a large list of standard puzzles with their first values, and the OEIS outputs whenever the sequence is already listed in the On-Line Encyclopedia of Integer Sequences [8]. Up to order 4 there are 114 sequences identified in OEIS, meaning that those sequences have already been found in various analytical, arithmetical or combinatorial contexts and can also be derived in our puzzle model, but 1339 are still unknown, that is, do have puzzle descriptions, but have not been encountered in other contexts.

2. Definition of standard puzzle

A piece is a square having four numbers, called labels, written in its corners. A puzzle is a connected arrangement of pieces in the ℤ × ℤ-plane such that the joining corners of all the pieces have the same labels (see Fig. 1). Pieces and puzzles can be translated, but can be neither rotated, nor reflected. The shape of each puzzle is a polyomino obtained from the puzzle by removing the labels.

A standard puzzle of shape λ is a puzzle such that the multi-set of all its labels is simply {1, 2, …, m}. This implies that m is the number of vertices of the polyomino λ. In particular, the four labels of a piece occurring in a standard puzzle are all distinct. Replacing the four

Figure 1. Adding a piece to a puzzle

labels by {1, 2, 3, 4} respecting the label ordering yields a standard piece. This operation is called a reduction and will be denoted by Ω (see Fig. 2). Two pieces are said to be identical if they have the same reductions. In the rest of the paper, puzzle means standard puzzle and piece means standard piece. We then have only twenty-four different pieces which are listed and coded in Table 1. Note that the letters “I” and “O” are not used.

Figure 2. Reduction of a piece

\begin{matrix}A = [\begin{matrix}4 3 \\ 1 2 \end{matrix}] & B = [\begin{matrix}3 4 \\ 1 2 \end{matrix}] & C = [\begin{matrix}4 2 \\ 1 3 \end{matrix}] & D = [\begin{matrix}2 4 \\ 1 3 \end{matrix}] & E = [\begin{matrix}3 2 \\ 1 4 \end{matrix}] & F = [\begin{matrix}2 3 \\ 1 4 \end{matrix}] \\ G = [\begin{matrix}4 3 \\ 2 1 \end{matrix}] & H = [\begin{matrix}3 4 \\ 2 1 \end{matrix}] & J = [\begin{matrix}4 1 \\ 2 3 \end{matrix}] & K = [\begin{matrix}1 4 \\ 2 3 \end{matrix}] & L = [\begin{matrix}3 1 \\ 2 4 \end{matrix}] & M = [\begin{matrix}1 3 \\ 2 4 \end{matrix}] \\ N = [\begin{matrix}4 2 \\ 3 1 \end{matrix}] & P = [\begin{matrix}2 4 \\ 3 1 \end{matrix}] & Q = [\begin{matrix}4 1 \\ 3 2 \end{matrix}] & R = [\begin{matrix}1 4 \\ 3 2 \end{matrix}] & S = [\begin{matrix}2 1 \\ 3 4 \end{matrix}] & T = [\begin{matrix}1 2 \\ 3 4 \end{matrix}] \\ U = [\begin{matrix}3 2 \\ 4 1 \end{matrix}] & V = [\begin{matrix}2 3 \\ 4 1 \end{matrix}] & W = [\begin{matrix}3 1 \\ 4 2 \end{matrix}] & X = [\begin{matrix}1 3 \\ 4 2 \end{matrix}] & Y = [\begin{matrix}2 1 \\ 4 3 \end{matrix}] & Z = [\begin{matrix}1 2 \\ 4 3 \end{matrix}] \end{matrix}

Table 1. The twenty-four pieces and its codes

Given a standard puzzle α, a set P of standard pieces is called support of the puzzle α if the reduction of each piece occurring in the puzzle α is an element of P. Hence, the minimal support of α is the set of all different pieces of α after reduction. For example, the standard puzzle given in Fig. 3 contains seven pieces, but only four of them are different [\begin{matrix}2 3 \\ 4 1 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}3 2 \\ 4 1 \end{matrix}]. The minimal support is the set of those four pieces and each set of pieces containing those four pieces is a support of the puzzle. The main problem discussed in the paper is the following.

Problem 1. Given a set of pieces P and a polyomino λ, count the number of standard puzzles of shape λ whose support is equal to P.

Figure 3. A standard puzzle

If the support is \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\} and the shape is the Ferrers diagram [15] of an integer

partition μ = (μ_{1} ≥ μ_{2} ≥ … ≥ μ_{ℓ} > 0) such that μ_{1} = μ_{2} and μ_{ℓ} ≥ 2, then a standard puzzle is just a standard Young tableau [14] of shape μ. Hence, the number of standard puzzles is the number of standard Young tableaux of fixed shape (See Fig. 4 in Section 4). Problem 1 can be solved by using the famous hook length formula [4].

However, the problem is very hard to solve in general. In the rest of the paper we focus our attention to a very special shape, namely, the 2 × n matrix. For convenience puzzles of shape 2 × n will be represented by two-row matrices. For example [\begin{matrix}5784 \\ 1236 \end{matrix}] stands for

In this case the set of all puzzles made by using the support P is denoted by P^{n}. For

example, take P = BC = \{B, C\} = \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}]\}. Then

BC^{2} = \{[\begin{matrix}34 \\ 12 \end{matrix}], [\begin{matrix}42 \\ 13 \end{matrix}]\}
BC^{3} = \{[\begin{matrix}563 \\ 124 \end{matrix}], [\begin{matrix}463 \\ 125 \end{matrix}], [\begin{matrix}456 \\ 123 \end{matrix}], [\begin{matrix}364 \\ 125 \end{matrix}], [\begin{matrix}356 \\ 124 \end{matrix}]\}
BC^{4} = \{[\begin{matrix}6784 \\ 1235 \end{matrix}], [\begin{matrix}5784 \\ 1236 \end{matrix}], [\begin{matrix}5684 \\ 1237 \end{matrix}], [\begin{matrix}5678 \\ 1234 \end{matrix}], [\begin{matrix}4785 \\ 1236 \end{matrix}], [\begin{matrix}4685 \\ 1237 \end{matrix}], [\begin{matrix}4678 \\ 1235 \end{matrix}],

[\begin{matrix}4586 \\ 1237 \end{matrix}], [\begin{matrix}4578 \\ 1236 \end{matrix}], [\begin{matrix}3785 \\ 1246 \end{matrix}], [\begin{matrix}3685 \\ 1247 \end{matrix}], [\begin{matrix}3678 \\ 1245 \end{matrix}], [\begin{matrix}3586 \\ 1247 \end{matrix}], [\begin{matrix}3578 \\ 1246 \end{matrix}]\}

The sequence (|BC^{n}|)_{n} = (|BC^{2}|, |BC^{3}|, |BC^{4}|, …), equal to

(2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, …),

is the well-known sequence of the Catalan numbers. However, many sequences defined by other supports are not identified in OEIS. For example, with P = CK = \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\} we have

CK^{2} = \{[\begin{matrix}42 \\ 13 \end{matrix}], [\begin{matrix}14 \\ 23 \end{matrix}]\}
CK^{3} = \{[\begin{matrix}625 \\ 134 \end{matrix}], [\begin{matrix}526 \\ 134 \end{matrix}], [\begin{matrix}426 \\ 135 \end{matrix}], [\begin{matrix}164 \\ 235 \end{matrix}]\}
CK^{4} = \{[\begin{matrix}8275 \\ 1346 \end{matrix}], [\begin{matrix}7285 \\ 1346 \end{matrix}], [\begin{matrix}6285 \\ 1347 \end{matrix}], [\begin{matrix}5286 \\ 1347 \end{matrix}], [\begin{matrix}4286 \\ 1357 \end{matrix}], [\begin{matrix}1847 \\ 2356 \end{matrix}], [\begin{matrix}1748 \\ 2356 \end{matrix}], [\begin{matrix}1648 \\ 2357 \end{matrix}]\}

The sequence (|CK^{n}|)_{n} is

(2, 4, 8, 26, 66, 276, 816, 4050, 13410, 75780, 274680 …)

which is not in OEIS.

3. The two-line matrix shape

When the shape is a 2 × n matrix, the enumeration problems for the supports P and P′ are equivalent, denoted by P ≡ P′, if P′ is obtained from P by applying the following basic transformations one or more times:

(T1) exchanging left column and right column in every piece;

(T2) exchanging top row and bottom row in every piece;

(T3) replacing each label a by (5 − a) in every piece.

For example, the support \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}]\} is equivalent to \{[\begin{matrix}3 4 \\ 2 1 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\} by (T1), to \{[\begin{matrix}1 2 \\ 4 3 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

by (T2) and to \{[\begin{matrix}1 2 \\ 4 3 \end{matrix}], [\begin{matrix}2 1 \\ 4 3 \end{matrix}]\} by (T3). The enumeration problems for the following supports P are all equivalent.

\{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}]\} ≡ \{[\begin{matrix}1 2 \\ 3 4 \end{matrix}], [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\} ≡ \{[\begin{matrix}4 3 \\ 2 1 \end{matrix}], [\begin{matrix}3 4 \\ 2 1 \end{matrix}]\} ≡ \{[\begin{matrix}2 1 \\ 3 4 \end{matrix}], [\begin{matrix}2 1 \\ 4 3 \end{matrix}]\}

≡ \{[\begin{matrix}2 1 \\ 4 3 \end{matrix}], [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\} ≡ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\} ≡ \{[\begin{matrix}2 1 \\ 3 4 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\} ≡ \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 2 1 \end{matrix}]\}.

A support P is said to be connected if for every pair of pieces from P, there is a puzzle containing those two pieces. The non-connected supports P can be fully characterized, and have one of the following three forms:

(a) \{[\begin{matrix}· & · \\ ∧ & ∧ \\ · & · \end{matrix}], [\begin{matrix}· & · \\ ∧ & ∧ \\ · & · \end{matrix}], ⋯, [\begin{matrix}· & · \\ ∧ & ∧ \\ · & · \end{matrix}]; [\begin{matrix}· & · \\ ∨ & ∨ \\ · & · \end{matrix}], [\begin{matrix}· & · \\ ∨ & ∨ \\ · & · \end{matrix}], ⋯, [\begin{matrix}· & · \\ ∨ & ∨ \\ · & · \end{matrix}]\}

(b)  \{[\begin{matrix}· & · \\ ∧ & ∨ \\ · & · \end{matrix}]; [\begin{matrix}· & · \\ ∧ & ∨ \\ · & · \end{matrix}]; ⋯; [\begin{matrix}· & · \\ ∧ & ∨ \\ · & · \end{matrix}]\}

(c)  \{[\begin{matrix}· & · \\ ∨ & ∧ \\ · & · \end{matrix}]; [\begin{matrix}· & · \\ ∨ & ∧ \\ · & · \end{matrix}]; ⋯; [\begin{matrix}· & · \\ ∨ & ∧ \\ · & · \end{matrix}]\}

For example, the support \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}]; [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\} is not connected (type (a)). It contains two connected components. For studying the enumeration problem we need only consider the connected supports. After reduction by (T1–T3), the number of possible connected supports P is shown in the following table

\begin{aligned}|P| & : & 1 & 2 & 3 & 4 & 5 & 6 & ⋯ \\ #\{P\} & : & 6 & 37 & 259 & 1391 & 5460 & ? & ⋯ \end{aligned}

The list of all those sets P with the sequences can be found in Section 6.

Problem 2. What is the sequence of the numbers of possible connected supports #{P} ?

Fix a support P. Let f_{n}[\begin{matrix}X \\ Y \end{matrix}] be the number of all puzzles of shape 2 × n such that the

rightmost column is [\begin{matrix}X \\ Y \end{matrix}]. Then, the number f_{n} of all puzzles of shape 2 × n is then

f_{n} = ∑_{X,Y} f_{n}[\begin{matrix}X \\ Y \end{matrix}],  (1)

where the sum ranges over all ordered pairs of [2n] × [2n].

The inverse reduction x′ = Ω^{−1}(x; X, Y) of x by {X, Y } is defined by

x′ = \begin{cases}x; & if x ≤ a − 1 \\ x + 1; & if a ≤ x ≤ b − 2 \\ x + 2, & if b − 1 ≤ x \end{cases}

where a = min(X, Y) and b = max(X, Y).

Proposition 1. Let 1 ≤ X, Y ≤ 2n and X ≠ Y. Then

f_{n}[\begin{matrix}X \\ Y \end{matrix}] = ∑_{x,y} f_{n−1}[\begin{matrix}x \\ y \end{matrix}]  (2)

where the sum ranges over all ordered pairs (x, y) of [2n−2]×[2n−2] such that the reduction

of [\begin{matrix}Ω^{−1}(x; X, Y) & X \\ Ω^{−1}(y; X, Y) & Y \end{matrix}] is a piece in P.

Formulas (1) and (2) are used for computing the first values of the number of standard puzzles (f_{n}).

4. Selected examples

In Section 6 we display the full list of all standard puzzle sequences including the outputs from OEIS if any. Each item in the dictionary may contain the following fields:

- Codes and pieces: The support P of the standard puzzles

- See also: List of other supports P′ such |P^{n}| = |P^{′n}|

- Seq: The sequence (|P^{n}|) for n = 2, 3, 4, …

- Var: This is a variant of the sequence of (|P^{n}|). Sometimes the sequence is not in OEIS but a slight modification of that sequence is in OEIS.

- OEIS: The OEIS output contains three parts: the OEIS code, the number of results found (written in square brackets [ ]), and the description of the first result found in OEIS.

In the present paper we will not reproduce all the formulas observed in the same way as formulas (3)–(10) that are presented further below. In fact, we have 114 sequences identified in OEIS up to order |P| = 4. It corresponds to 309 supports P, that is, we have 309 formulas to prove!

For example, we find the following item in the dictionary.

BC \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}]\} See also BD

Seq= 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012 OEIS: \underline{A000108} [5] Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

From the above item BC in the dictionary we extract the following results.

Theorem 2. We have

|BC^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}]\}^{n}| = C_{n};  (3)
|BD^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\}^{n}| = C_{n},  (4)

where C_{n} = \frac{1}{n+1} \binom{2n}{n} is the Catalan numbers [10].

Proof. Equation (4) is well-known since the puzzles with support BD = \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\} are just the standard Young tableaux, which can be counted by the famous hook length formula [4]. In fact, each puzzle with support BD is just a labelling of the vertices of the following diagram such that the labels are increasing in the sense of the arrows [9, p. 227].

Figure 4. Puzzles having the support BD

However, identity (3) cannot be directly found in the references. A further bijection is to be constructed to map a puzzle from BC^{n} onto a puzzle from BD^{n}. Each puzzle with support BC contains no piece C, or only one piece C at the rightmost position. Hence, the puzzles with support BC can be characterized by the following two diagrams:

Figure 5. Puzzles having the support BC

In Fig. 4 and 5 the labels of certain vertices can only have fixed values, in particular maximal or minimal values. Removing those vertices yields the simplified diagrams Figures 6 and 7. The standard labellings of the diagram in Fig. 6 such that a < b (respectively a > b) are in bijection with the standard labellings of the left diagram (respectively right diagram) in Fig. 7. This gives a bijection between BC^{n} and BD^{n}. □

Figure 6. Puzzles having the support BD (after simplification)

Figure 7. Puzzles having the support BC (after simplification)

Recall that the classical tangent numbers T_{2n+1}, secant numbers E_{2n} and the unsigned Genocchi numbers G_{2n} are defined by the following generating functions [13].

tan u = ∑_{n≥0} \frac{u^{2n+1}}{(2n + 1)!}T_{2n+1}

sec u = ∑_{n≥0} \frac{u^{2n}}{(2n)!}E_{2n}

\frac{2u}{e^{u} + 1} = u + ∑_{n≥1}(−1)^{n} \frac{u^{2n}}{(2n)!}G_{2n}

It is well-known that T_{2n+1} and E_{2n} count the numbers of alternating permutations of length 2n + 1 and 2n respectively [13]. The unsigned Genocchi numbers G_{2n} = nT_{2n−1}/2^{2n−2} count the numbers of surjective staircases [1]. The support P = BCEG involves those numbers.

BCEG \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\} See also BEGJ, BEGY, BEJV, BEVY, BJTV, EJRV

Seq= 4, 24, 272, 4960, 132672, 4893056, 237969664, 14756156928 Var= 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905 OEIS: \underline{A110501} [2] Unsigned Genocchi numbers (of first kind) of even index.

From the above item BCEG in the dictionary we read the following seven identities as shown in Theorem 3 and Conjecture 4.

Theorem 3. We have

|BEGJ^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} ;  (5)
|BEGY^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}], [\begin{matrix}2 1 \\ 4 3 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} ;  (6)
|BEJV^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} ;  (7)
|BEV Y^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}], [\begin{matrix}2 1 \\ 4 3 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} ;  (8)
|BJT V^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} ;  (9)
|EJRV^{n}| = |\{[\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} .  (10)

Proof. The enumeration of |BJT V^{n}| is derived in [5, 2] and identity (9) is proven to be true. Notice that no easy direct proof of (9) is known. We prove that all the six left-hand sides of equations (5)-(10) are equal to

|BGT Y^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}], [\begin{matrix}2 1 \\ 4 3 \end{matrix}]\}^{n}| .  (11)

Let S be a subset of {1, 2, …, n}. The flip map ϕ_{S} : α ↦ β is a transformation which

maps a puzzle α = [\begin{matrix}x_{1}x_{2} ⋯ x_{n} \\ y_{1}y_{2} ⋯ y_{n} \end{matrix}] onto β = [\begin{matrix}a_{1}a_{2} ⋯ a_{n} \\ b_{1}b_{2} ⋯ b_{n} \end{matrix}] such that a_{i} = x_{i}, b_{i} = y_{i} for i ∉ S and

a_{i} = y_{i}, b_{i} = x_{i} for i ∈ S. Notice that the flip map is an involution. The following diagram shows the actions of the flip maps onto some pieces:

(B, G, T, Y) ⟶^{ϕ_{{1,2}}} (T, Y, B, G)
↓ ϕ_{{2}}
(E, J, R, V) ⟶^{ϕ_{{1,2}}} (R, V, E, J)

The important facts are that the flip map ϕ_{{1,2}} does not change the two supports {B, G, T, Y } and {E, J, R, V }; moreover, those two supports do not contain any common piece. We can prove that all the 2^{4} = 16 supports P derived from the Cartesian product Γ := {B, E} × {G, J} × {T, R} × {Y, V } yield the same enumeration sequences (|P^{n}|). For example, we will explain why

|BGT Y^{n}| = |BJRY^{n}| .  (12)

Notice that

|BGT Y^{n}| = |BG^{n}| + |T Y^{n}| = 2|BG^{n}| .  (13)

Let α be a puzzle in BGT Y^{n}. We construct a puzzle β in BJRY^{n} in a unique manner by applying an adequate sequence of flip maps.

Start at γ := α; from left to right look for the first piece in γ that is not in {B, J, R, Y }, that is, the first piece equal to G or T. Let (i, i + 1) be the position of that piece. Apply the flip map ϕ_{{i+1,i+2,...,n−1}} to γ, to obtain a new puzzle. By convention, let γ := ϕ_{{i+1,i+2,...,n−1}}(γ). Hence, the piece at position (i, i + 1) becomes J or R, and all pieces on the right of that position are still in {B, G, T, Y }, by the important facts mentioned above. Repeat this process until no more G or T are in γ to get the puzzle β := γ in BJRY^{n}.

For example, take α = [\begin{matrix}11 & 13 & 10 & 6 & 12 & 8 & 14 \\ 7 & 9 & 2 & 1 & 4 & 3 & 5 \end{matrix}] ∈ BGT Y^{n}. The calculation in Table 4

shows that β = [\begin{matrix}11 & 13 & 2 & 1 & 12 & 3 & 14 \\ 7 & 9 & 10 & 6 & 4 & 8 & 5 \end{matrix}] ∈ BJRY^{n}. Note that the number of a piece, for example B, common to two different supports, is not preserved.

This process is reversible due to the important facts mentioned above. Hence, identity (12) is proved. Now, since BJT V, EJRV ∈ Γ, identities (9) and (10) are proved. In the same manner, BGRV, EGT Y, BJRV, BJRY ∈ Γ and BGRV ≡ BEGJ, EGT Y ≡ BEGY, BJRV ≡ BEJV, EJRY ≡ BEV Y. This achieves the proof of (5)–(8). □

The next identity cannot be proved in the same manner, since the flip maps are not enough to produce the bijection. Further operations are to be constructed.

Conjecture 4. We have

|BCEG^{n}| = |\{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\}^{n}| = nT_{2n−1}/2^{n−2} .  (14)

puzzle  first G/T position  flip
→ [\begin{matrix}11 & & 13 & & 10 & & 6 & & 12 & & 8 & & 14 \\ & B & & G & & G & & B & & G & & B \\ 7 & & 9 & & 2 & & 1 & & 4 & & 3 & & 5 \end{matrix}]  G  (2, 3) S = \{3, 4, 5, 6, 7\}

→ [\begin{matrix}11 & & 13 & & 2 & & 1 & & 4 & & 3 & & 5 \\ & B & & J & & Y & & T & & Y & & T \\ 7 & & 9 & & 10 & & 6 & & 12 & & 8 & & 14 \end{matrix}]  T  (4, 5)  S = \{5, 6, 7\}

→ [\begin{matrix}11 & & 13 & & 2 & & 1 & & 12 & & 8 & & 14 \\ & B & & J & & Y & & R & & G & & B \\ 7 & & 9 & & 10 & & 6 & & 4 & & 3 & & 5 \end{matrix}]  G  (5, 6)  S = \{6, 7\}

→ [\begin{matrix}11 & & 13 & & 2 & & 1 & & 12 & & 3 & & 5 \\ & B & & J & & Y & & R & & J & & T \\ 7 & & 9 & & 10 & & 6 & & 4 & & 8 & & 14 \end{matrix}]  T  (6, 7)  S = \{7\}

→ [\begin{matrix}11 & & 13 & & 2 & & 1 & & 12 & & 3 & & 14 \\ & B & & J & & Y & & R & & J & & R \\ 7 & & 9 & & 10 & & 6 & & 4 & & 8 & & 5 \end{matrix}]

Table 2. Bijection from BGT Y^{n} onto BJRY^{n}

Other supports P giving classical numbers are listed below without proof.

ACX: Fibonacci numbers; AB, ABCD: double factorial numbers; CDW: Koch snowflake, number of angles after n iterations; ACET: number of branching configurations of RNA with n or fewer hairpins; ACMT: super-Catalan numbers or little Schröder numbers; ADHN: number of permutations of length 2n − 1 with no local maxima or minima in even positions; AELM: number of permutations of length n which avoid the patterns 231, 12534; BDFK: number of Dyck paths of semilength n having no DUDU’s starting at level 1.

5. Where are the secant numbers?

Since the tangent numbers appear in the puzzle sequence with support BEGJ (see (5)), a natural question rises: what about the secant numbers (E_{2n})_{n≥2}:

(5, 61, 1385, 50521, 2702765, 199360981, 19391512145, …) ?  (15)

The initial motivation was to define a support that would generate a puzzle sequence for the secant numbers. ^{1} We developed a computer program to generate the puzzle sequences up to order |P| = 4. Unfortunately, no secant number sequence could be found.

1 A support, namely, P = ABDGH, that generates the secant numbers is found recently by Jiaxi Lu and Yuanzhe Ding [7].

Notice that the secant numbers appeared in the puzzle sequence for the support BJT V, but a non-trivial modification is required. By (9) we have

|BJT V^{n}| = ∑_{X,Y} |BJT V^{n}[\begin{matrix}X \\ Y \end{matrix}]| = \frac{n}{2^{n−2}} T_{2n−1},  (16)

where BJT V^{n}[\begin{matrix}X \\ Y \end{matrix}] is the subset of puzzles in NJT V^{n} such that the rightmost column is [\begin{matrix}X \\ Y \end{matrix}].

In [3] we obtained the following identity by using the signed doubloon model:

∑_{X,Y} |BJT V^{n}[\begin{matrix}X \\ Y \end{matrix}]| × Q_{n}(X, Y) = 2^{−n}E_{2n},  (17)

where

Q_{n}(X, Y) = \begin{cases}∑{Y −1 \atop k=X} \binom{2n}{k} & if X < Y \\ 0 & if X > Y \end{cases}

However, the secant number sequence (15) does not directly appear in the dictionary without the coefficient Q_{n}(X, Y). It is possible that order |P| ≤ 4 is not large enough. Why do not choose a bigger support, for example P = CEHJLP RV X. The puzzle sequence is

(9, 111, 2505, 91961, 4913789, 364074545, 35418898477, …)  (18)

We do not obtain the secant sequence (15). It is amusing that the quotient of (18) by the secant sequence (15) is approximately 1.8:

(1.800, 1.820, 1.809, 1.820, 1.818, 1.826, 1.827, …)

What does it mean?

6. Dictionary of the standard puzzle sequences

Sequence identified for |P| = 1

A \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}]\} See also D

Seq= 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 OEIS: \underline{A000012} [1450] The simplest sequence of positive numbers: the all 1’s sequence.

B \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}]\}

Seq= 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786 OEIS: \underline{A000108} [5] Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

C \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}]\} See also E, F

Seq= 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 OEIS: \underline{A121373} [1470] Expansion of f(q) = f(q, -qˆ 2) in powers of q where f(q,r) is the Ramanujan two variable theta function.

Sequence identified for |P| = 2

AB \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}]\}

Seq= 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200 OEIS: \underline{A000165} [1] Double factorial numbers: (2n)!! = 2ˆ n*n!.

AC \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}]\} See also AK, AX, CD, CX, DE, DF, EK, FK

Seq= 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 OEIS: \underline{A055642} [327] Number of digits in decimal expansion of n.

AD \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\}

Seq= 2, 6, 23, 106, 567, 3434, 23137, 171174, 1376525, 11934581, 110817423 OEIS: \underline{A125273} [1] Eigensequence of triangle A085478: a(n) = Sum \{k=0..n-1\} A085478(n-1,k)*a(k) for n>0 with a(0)=1.

AE \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}]\}

Seq= 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 OEIS: \underline{A000027} [618] The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

AF \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\}

Seq= 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67 OEIS: \underline{A000124} [6] Central polygonal numbers (the Lazy Caterer’s sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pa...

AL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\}

Seq= 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122 OEIS: \underline{A002522} [2] nˆ 2 + 1.

AP \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}]\} See also DQ

Seq= 2, 7, 16, 29, 46, 67, 92, 121, 154, 191, 232 OEIS: \underline{A130883} [1] 2nˆ 2-n+1.

AR \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}]\} See also CM, DJ

Seq= 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 OEIS: \underline{A005843} [25] The even numbers: a(n) = 2n.

BC \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}]\} See also BD

Seq= 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012 OEIS: \underline{A000108} [5] Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

12 GUO-NIU HAN

BE \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}]\} See also ER

Seq= 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572 OEIS: \underline{A068875} [2] Expansion of (1+x*C)*C, where C = (1-(1-4*x)ˆ (1/2))/(2*x) is g.f. for Catalan numbers, A000108.

BF \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\}

Seq= 2, 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582 OEIS: \underline{A005807} [2] Sum of adjacent Catalan numbers.

BG \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\}

Seq= 2, 12, 136, 2480, 66336, 2446528, 118984832, 7378078464, 568142287360 OEIS: \underline{A117513} [1] Number of ways of arranging 2n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair o...

BJ \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\}

Seq= 2, 6, 22, 84, 324, 1254, 4862, 18876, 73372, 285532, 1112412 OEIS: \underline{A121686} [1] Number of branches in all binary trees with n edges. A binary tree is a rooted tree in which each vertex has at most two chi...

BL \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\}

Seq= 2, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512 OEIS: \underline{A028329} [1] Twice central binomial coefficients.

Sequence unknown for |P| = 2

AH \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 2 1 \end{matrix}]\}

Seq= 2, 12, 132, 2372, 62304, 2261668, 108184432, 6600715188, 500046044352

BN \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 3 1 \end{matrix}]\}

Seq= 2, 9, 74, 974, 18831, 502459, 17671764, 792391014, 44129928926, 2987912108763

BQ \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 1 \\ 3 2 \end{matrix}]\}

Seq= 2, 5, 21, 96, 440, 1989, 8855, 38896, 168948, 727090, 3105322

CK \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\}

Seq= 2, 4, 8, 26, 66, 276, 816, 4050, 13410, 75780, 274680

CP \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}]\}

Seq= 2, 10, 82, 1162, 23026, 657148, 23719394, 1137763610, 65032729314

CR \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}]\}

Seq= 2, 5, 18, 91, 563, 4299, 37686, 384543, 4357567, 55614775, 772479331

DN \{[\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}4 2 \\ 3 1 \end{matrix}]\}

Seq= 2, 14, 168, 3352, 96816, 3875904, 204185344, 13726330128, 1145508631264

ENUMERATION OF STANDARD PUZZLES 13

EU \{[\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 2 \\ 4 1 \end{matrix}]\}

Seq= 2, 6, 26, 150, 1230, 10038, 125490, 1292166, 22184550, 271843110, 6022023210

EV \{[\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\}

Seq= 2, 8, 58, 712, 12564, 310256, 10025978, 415159208, 21288518044, 1329526717840

FU \{[\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 2 \\ 4 1 \end{matrix}]\}

Seq= 2, 12, 106, 1680, 37434, 1171968, 48008850, 2516016384, 163509808050

Sequence identified for |P| = 3

ABC \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}]\} See also ABE, ABAFT

Seq= 3, 12, 72, 576, 5760, 69120, 967680, 15482880, 278691840, 5573836800 OEIS: \underline{A052676} [1] A simple regular expression in a labeled universe.

ABD \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\}

Seq= 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575 OEIS: \underline{A001147} [1] Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).

ABK \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\}

Seq= 3, 10, 56, 432, 4224, 49920, 691200, 10967040, 196116480, 3901685760 Var= 3, 10, 28, 72, 176, 416, 960, 2176, 4864, 10752, 23552 OEIS: \underline{A128135} [1] Row sums of A128134. ACE \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}]\} See also ACZ, AKM, AKS, AMX, DEM, DFM Seq= 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 OEIS: \underline{A000027} [527] The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

ACF \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\}

Seq= 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68 OEIS: \underline{A022856} [4] a(n) = n-2 + Sum of a(i+1)mod(a(i)) for i = 1 to n-2, for n ≥ 3.

ACL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\} See also AMR, CDM

Seq= 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123 OEIS: \underline{A059100} [4] nˆ 2+2.

ACM \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\} See also AFM, AFZ

Seq= 3, 6, 14, 35, 90, 234, 611, 1598, 4182, 10947, 28658 OEIS: \underline{A032908} [2] One of 4 3rd-order recurring sequences for which the first derived sequence and the Galois transformed sequence coincide.

ACS \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\} See also ATX

Seq= 3, 7, 21, 71, 253, 925, 3433, 12871, 48621, 184757, 705433 Var= 0, 4, 18, 68, 250, 922, 3430, 12868, 48618, 184754, 705430 OEIS: \underline{A115112} [1] Number of different ways to select n elements from two sets of n elements under the precondition of choosing at least one el...

ACT \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

Seq= 3, 6, 17, 62, 259, 1162, 5441, 26234, 129283, 648142, 3294865 Var= 2, 5, 16, 61, 258, 1161, 5440, 26233, 129282, 648141, 3294864 OEIS: \underline{A104858} [1] Partial sums of the little Schroeder numbers (A001003).

ACX \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 3 \\ 4 2 \end{matrix}]\} See also DEK, DFK

Seq= 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 OEIS: \underline{A000045} [15] Fibonacci numbers: F(n) = F(n-1) + F(n-2), F(0) = 0, F(1) = 1, F(2) = 1, ...

AEF \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\} See also AEQ

Seq= 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78 OEIS: \underline{A000217} [29] Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.

AEJ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\} See also AKR, AKV, ARX, CDJ, CDL, CEM, CFM, DEJ, DFJ Seq= 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 OEIS: \underline{A005408} [34] The odd numbers: a(n) = 2n+1.

AEL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\} See also AFQ

Seq= 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133 OEIS: \underline{A002061} [2] Central polygonal numbers: nˆ 2 - n + 1.

AEM \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 3, 5, 10, 23, 57, 146, 379, 989, 2586, 6767, 17713 Var= 0, 2, 7, 20, 54, 143, 376, 986, 2583, 6764, 17710 OEIS: \underline{A035508} [2] Fibonacci(2n+2)-1.

AES \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\}

Seq= 3, 6, 17, 58, 212, 794, 3005, 11442, 43760, 167962, 646648 Var= 1, 4, 15, 56, 210, 792, 3003, 11440, 43758, 167960, 646646 OEIS: \underline{A001791} [2] Binomial coefficients C(2n,n-1).

AEW \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\}

Seq= 3, 7, 22, 73, 237, 746, 2287, 6867, 20286, 59157, 170713 Var= 1, 5, 20, 71, 235, 744, 2285, 6865, 20284, 59155, 170711 OEIS: \underline{A054444} [1] Even indexed members of A001629(n), n ≥ 2, (Fibonacci convolution).

AEZ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\}

Seq= 3, 7, 19, 52, 140, 372, 981, 2577, 6757, 17702, 46358 Var= 0, 4, 16, 49, 137, 369, 978, 2574, 6754, 17699, 46355 OEIS: \underline{A114185} [1] F(2n)-n-1, where F(n)=Fibonacci number.

AFL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\}

Seq= 3, 8, 16, 27, 41, 58, 78, 101, 127, 156, 188 OEIS: \underline{A104249} [1] (3*nˆ 2+n+2)/2.

AFS \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\}

Seq= 3, 8, 31, 132, 564, 2382, 9951, 41228, 169768, 695862, 2842228 Var= 1, 6, 29, 130, 562, 2380, 9949, 41226, 169766, 695860, 2842226 OEIS: \underline{A008549} [1] Number of ways of choosing at most n-1 items from a set of size 2n+1.

AKP \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}]\} See also APX, CDQ, DEQ, DFQ

Seq= 3, 8, 17, 30, 47, 68, 93, 122, 155, 192, 233 OEIS: \underline{A033816} [1] 2nˆ 2 + 3n + 3.

AKT \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\} See also BFM

Seq= 3, 5, 10, 24, 66, 198, 627, 2057, 6919, 23715, 82501 OEIS: \underline{A155587} [1] Expansion of (1+x*c(x))/(1-x), c(x) the g.f. of A000108.

AKU \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}3 2 \\ 4 1 \end{matrix}]\} See also AKX, CDE, CDF, DEF

Seq= 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 OEIS: \underline{A010701} [202] Constant sequence.

AKW \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\} See also APS, DFW

Seq= 3, 9, 33, 129, 513, 2049, 8193, 32769, 131073, 524289, 2097153 OEIS: \underline{A084508} [2] Partial sums of A084509. Positions of ones in the first differences of A084506.

AMV \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\} See also ARS

Seq= 3, 8, 31, 126, 509, 2044, 8187, 32762, 131065, 524280, 2097143 Var= 0, 5, 28, 123, 506, 2041, 8184, 32759, 131062, 524277, 2097140 OEIS: \underline{A124133} [1] a(n)=(-1/2)*sum \{i1+i2+i3=2n\} ((2*n)!/(i1! i2! i3!))*B(i1) where B are the Bernoulli numbers.

APR \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}]\} See also CFW, DJQ

Seq= 3, 10, 21, 36, 55, 78, 105, 136, 171, 210, 253 OEIS: \underline{A014105} [1] Second hexagonal numbers: n(2n+1).

ART \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\} See also BET

Seq= 3, 8, 25, 84, 294, 1056, 3861, 14300, 53482, 201552, 764218 OEIS: \underline{A038665} [1] Convolution of A007054 (super ballot numbers) with A000984 (central binomial coefficients).

ARV \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\} See also CEW

Seq= 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43 OEIS: \underline{A004767} [7] 4n+3.

BCD \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\}

Seq= 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888 OEIS: \underline{A000245} [3] 3(2n)!/((n+2)!(n-1)!).

BCE \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}]\} See also BDF, BEK

Seq= 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582, 266798 OEIS: \underline{A005807} [3] Sum of adjacent Catalan numbers.

BDE \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}]\}

Seq= 3, 8, 23, 70, 222, 726, 2431, 8294, 28730, 100776, 357238 OEIS: \underline{A000782} [2] 2*Catalan(n)-Catalan(n-1).

BDL \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\}

Seq= 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078 OEIS: \underline{A001700} [3] C(2n+1, n+1): number of ways to put n+1 indistinguishable balls into n+1 distinguishable boxes = number of (n+1)-st degree m...

BEM \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 3, 6, 14, 37, 107, 329, 1055, 3486, 11780, 40510, 141286 OEIS: \underline{A081293} [1] a(n) = A000108(n)+A014137(n).

BFL \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\}

Seq= 3, 5, 14, 45, 154, 546, 1980, 7293, 27170, 102102, 386308 OEIS: \underline{A078718} [1] Let f(i,j) = Sum(binom(2*i,k)*binom(2*j,i+j-k)*(-1)ˆ (i+j-k),k=0..2*i) (this is essentially the same as the triangle in...

BFT \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

Seq= 3, 6, 15, 42, 126, 396, 1287, 4290, 14586, 50388, 176358 OEIS: \underline{A120589} [1] Self-convolution of A120588, such that a(n) = 3*A120588(n) for n≥2.

BLM \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 3, 6, 16, 51, 177, 639, 2355, 8790, 33100, 125478, 478194 Var= 1, 4, 14, 49, 175, 637, 2353, 8788, 33098, 125476, 478192 OEIS: \underline{A079309} [1] a(n) = C(1,1)+C(3,2)+C(5,3)+...+C(2n-1,n).

CDW \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\} See also DEW

Seq= 3, 6, 18, 66, 258, 1026, 4098, 16386, 65538, 262146, 1048578 OEIS: \underline{A178789} [1] Koch snowflake: number of angles after n iterations.

ENUMERATION OF STANDARD PUZZLES 17

DJM \{[\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 3, 8, 21, 46, 87, 148, 233, 346, 491, 672, 893 OEIS: \underline{A179903} [1] (1, 3, 5, 7, 9...) convolved with (1, 0, 3, 5, 7, 9,...)

EFK \{[\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\}

Seq= 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96 OEIS: \underline{A029744} [8] Numbers of the form 2ˆ n or 3*2ˆ n.

Sequence unknown for |P| = 3 (partial)

ABG \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\}

Seq= 3, 23, 327, 7465, 249885, 11532671, 701867995, 54461600179, 5247921916235

ABH \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 2 1 \end{matrix}]\}

Seq= 3, 24, 345, 7920, 264873, 12190108, 739050425, 57109234080, 5479466654645

ABJ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\} See also ABL

Seq= 3, 14, 96, 858, 9420, 122490, 1839600, 31325490, 596291220, 12546094050

ABN \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 3 1 \end{matrix}]\}

Seq= 3, 19, 217, 3985, 107547, 4001027, 196224625, 12270923649, 953000374835

ABP \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}]\}

Seq= 3, 17, 116, 1048, 11712, 155520, 2388480, 41610240, 810270720, 17433722880

ABQ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 1 \\ 3 2 \end{matrix}]\}

Seq= 3, 14, 104, 1020, 12264, 173580, 2818080, 51535260, 1047274200, 23400192060

⋮

Sequence identified for |P| = 4

ABCD \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}]\} See also ABDE, ABDF

Seq= 4, 20, 140, 1260, 13860, 180180, 2702700, 45945900, 872972100, 18332414100 Var= 1, 5, 35, 315, 3465, 45045, 675675, 11486475, 218243025, 4583103525 OEIS: \underline{A051577} [2] (2*n+3)!!/3, related to A001147 (odd double factorials). ABCE \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}]\} See also ABCF, ABCM, ABCT, ABEF, ABEZ, ABRX, ABRZ, ABTX, ACER, ACRT, AERX, AERZ, AETX, BCEX, BCRX, BCTX, CERX Seq= 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400 OEIS: \underline{A032184} [2] ”CIJ” (necklace, indistinct, labeled) transform of 1,3,5,7...

ABCJ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\} See also ABCK, ABCL, ABDK, ABEJ, ABEL, ABFJ, ABFL

Seq= 4, 18, 120, 1050, 11340, 145530, 2162160, 36486450, 689188500, 14404039650 Var= 2, 6, 30, 210, 1890, 20790, 270270, 4054050, 68918850, 1309458150 OEIS: \underline{A097801} [2] (2*n)!/(n!*2ˆ (n-1)).

ABCZ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\} See also ABEM, ABFM, ABFZ

Seq= 4, 14, 80, 634, 6332, 75974, 1063624, 17017970, 306323444, 6126468862 Var= 3, 13, 79, 633, 6331, 75973, 1063623, 17017969, 306323443, 6126468861 OEIS: \underline{A010844} [1] a(n) = 2*n*a(n-1) + 1 with a(0)=1.

ABDJ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\} See also ABDL

Seq= 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600, 40874803200 OEIS: \underline{A002866} [3] a(0) = 1; for n>0, a(n) = 2ˆ (n-1)*n!.

ABDP \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}]\} See also ABDU, ABDV

Seq= 4, 25, 210, 2205, 27720, 405405, 6756750, 126351225, 2618916300, 59580345825 Var= 1, 5, 35, 315, 3465, 45045, 675675, 11486475, 218243025, 4583103525 OEIS: \underline{A051577} [2] (2*n+3)!!/3, related to A001147 (odd double factorials).

ABKM \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\} See also ABKS

Seq= 4, 12, 60, 444, 4284, 50364, 695484, 11017404, 196811964, 3912703164 OEIS: \underline{A004400} [1] 1 + Sum 2ˆ k k!, k = 1 . . n.

ABMX \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}1 3 \\ 4 2 \end{matrix}]\}

Seq= 4, 13, 73, 577, 5761, 69121, 967681, 15482881, 278691841, 5573836801 Var= 3, 12, 72, 576, 5760, 69120, 967680, 15482880, 278691840, 5573836800 OEIS: \underline{A052676} [1] A simple regular expression in a labeled universe.

ACDF \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\} See also ACDM, ACFK, ACKM, ADFM, ADFZ, ADKX, ADKZ, ADMX, AFKX, AFKZ, AFMX, CDFX, CDKX, CDMX, CFKX

Seq= 4, 12, 46, 212, 1134, 6868, 46274, 342348, 2753050, 23869162, 221634846 Var= 2, 6, 23, 106, 567, 3434, 23137, 171174, 1376525, 11934581, 110817423 OEIS: \underline{A125273} [1] Eigensequence of triangle A085478: a(n) = Sum \{k=0..n-1\} A085478(n-1,k)*a(k) for n>0 with a(0)=1.

ACEF \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\} See also ACEQ

Seq= 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79 OEIS: \underline{A000124} [7] Central polygonal numbers (the Lazy Caterer’s sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pa...

ACEJ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\} See also AKMX, AKRU, AKRX, AKSU, AKVX, CDEJ, CDEL, CDFJ, CDFL, CEFM, DEFJ, DEFM

Seq= 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 OEIS: \underline{A005843} [41] The even numbers: a(n) = 2n.

ACEL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\} See also ACFQ, AKMR, AMRX, CDEM, CDFM

Seq= 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134 OEIS: \underline{A014206} [2] nˆ 2+n+2.

ACEM \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\} See also ACEZ, AEFM

Seq= 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369 OEIS: \underline{A055588} [2] a(n)=3a(n-1)-a(n-2)-1; a(0)=1, a(1)=2.

ACES \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\} See also AMTX, BCFM

Seq= 4, 11, 36, 127, 463, 1717, 6436, 24311, 92379, 352717, 1352079 OEIS: \underline{A112849} [1] Number of congruence classes (epimorphisms/vertex partitionings induced by graph endomorphisms) of undirected cycles of even...

ACET \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

Seq= 4, 10, 32, 122, 516, 2322, 10880, 52466, 258564, 1296282, 6589728 OEIS: \underline{A176006} [1] The number of branching configurations of RNA (see Sankoff, 1985) with n or fewer hairpins.

ACEW \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\} See also AEQW

Seq= 4, 11, 34, 106, 325, 978, 2896, 8463, 24466, 70102, 199369 Var= 3, 10, 33, 105, 324, 977, 2895, 8462, 24465, 70101, 199368 OEIS: \underline{A027989} [1] a(n) = self-convolution of row n of array T given by A027926.

ACFL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\}

Seq= 4, 9, 17, 28, 42, 59, 79, 102, 128, 157, 189 Var= 0, 5, 13, 24, 38, 55, 75, 98, 124, 153, 185 OEIS: \underline{A140090} [1] n(3n+7)/2.

ACFM \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\} See also AFQZ

Seq= 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314 OEIS: \underline{A052995} [3] A simple regular expression.

ACLS \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\} See also ARTX, BCEM, BDET, BDFL

Seq= 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512, 1410864 OEIS: \underline{A100320} [2] A Catalan transform of (1+2x)/(1-2x).

ACMT \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

Seq= 4, 12, 46, 198, 904, 4280, 20794, 103050, 518860, 2646724, 13648870 Var= 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869 OEIS: \underline{A001003} [2] Schroeder’s second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

ACXZ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 3 \\ 4 2 \end{matrix}], [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\} See also DEFK, DEKM, DFKM

Seq= 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 OEIS: \underline{A000079} [30] Powers of 2: a(n) = 2ˆ n.

ADHN \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 4 \\ 2 1 \end{matrix}], [\begin{matrix}4 2 \\ 3 1 \end{matrix}]\}

Seq= 4, 42, 816, 25520, 1170240, 73992912, 6169370368, 655847011584 Var= 2, 14, 204, 5104, 195040, 10570416, 771171296, 72871890176, 8658173200896 OEIS: \underline{A122647} [1] Number of permutations of length 2n-1 with no local maxima or minima in even positions.

AEFJ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}]\}

Seq= 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89 OEIS: \underline{A034856} [1] C(n + 1, 2) + n - 1.

AEFL \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}]\} See also AELQ

Seq= 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199 OEIS: \underline{A005448} [1] Centered triangular numbers: 3n(n-1)/2 + 1.

AEFQ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 3 2 \end{matrix}]\} See also AEJL

Seq= 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 OEIS: \underline{A000290} [9] The squares: a(n) = nˆ 2.

AEFS \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\} See also AEQS

Seq= 4, 12, 46, 188, 774, 3174, 12954, 52668, 213526, 863822, 3488874 Var= 2, 10, 44, 186, 772, 3172, 12952, 52666, 213524, 863820, 3488872 OEIS: \underline{A068551} [1] 4ˆ n - binomial(2n,n).

AEFZ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 2 \\ 4 3 \end{matrix}]\} See also AEQZ

Seq= 4, 11, 31, 85, 228, 604, 1590, 4173, 10937, 28647, 75014 Var= 2, 9, 29, 83, 226, 602, 1588, 4171, 10935, 28645, 75012 OEIS: \underline{A152891} [1] a=b=0;b(n)=b+n+a;a(n)=a+n+b.

AEJS \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}]\}

Seq= 4, 10, 32, 114, 422, 1586, 6008, 22882, 87518, 335922, 1293294 Var= 2, 8, 30, 112, 420, 1584, 6006, 22880, 87516, 335920, 1293292 OEIS: \underline{A162551} [1] 2 * C(2n,n-1).

AELM \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 10, 28, 82, 242, 710, 2064, 5946, 16992, 48222, 136034 Var= 2, 5, 14, 41, 121, 355, 1032, 2973, 8496, 24111, 68017 OEIS: \underline{A116845} [1] Number of permutations of length n which avoid the patterns 231, 12534.

AELW \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\}

Seq= 4, 12, 40, 132, 422, 1310, 3972, 11824, 34692, 100612, 289034 Var= 3, 11, 39, 131, 421, 1309, 3971, 11823, 34691, 100611, 289033 OEIS: \underline{A166336} [1] Expansion of (1-4x+7xˆ 2-4xˆ 3+xˆ 4)/(1-7x+17xˆ 2-17xˆ 3+7xˆ 4-xˆ 5)

AFLQ \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}4 1 \\ 3 2 \end{matrix}]\} See also AKPR, AKPV, APRX, CDJQ, CDLQ, CEMQ, CFMQ, DEJQ, DFJQ Seq= 4, 11, 22, 37, 56, 79, 106, 137, 172, 211, 254 OEIS: \underline{A084849} [1] 1+n+2nˆ 2.

AKMP \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}]\} See also AMPX

Seq= 4, 13, 56, 241, 1000, 4061, 16336, 65473, 262064, 1048477, 4194184 Var= 3, 12, 55, 240, 999, 4060, 16335, 65472, 262063, 1048476, 4194183 OEIS: \underline{A024038} [1] 4ˆ n-nˆ 2.

AKMT \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\} See also BDFM

Seq= 4, 9, 23, 65, 197, 626, 2056, 6918, 23714, 82500, 290512 OEIS: \underline{A014137} [1] Partial sums of Catalan numbers (A000108).

AKMV \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\} See also AKRS, AKSV, AMVX, CDEW, CDJW, CDLW, DEJW Seq= 4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154 Var= 2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577 OEIS: \underline{A052539} [2] 4ˆ n + 1.

AKPU \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}], [\begin{matrix}3 2 \\ 4 1 \end{matrix}]\} See also AKPX, CDEQ, CDFQ, DEFQ

Seq= 4, 9, 18, 31, 48, 69, 94, 123, 156, 193, 234 OEIS: \underline{A100037} [1] Positions of occurrences of the natural numbers as second subsequence in A100035.

AKRV \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\} See also ARVX, CDJL, CEJM, CEQW

Seq= 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44 OEIS: \underline{A008586} [6] Multiples of 4.

AKRW \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\} See also AMPR, APRS

Seq= 4, 15, 62, 253, 1020, 4091, 16378, 65529, 262136, 1048567, 4194294 Var= 3, 14, 61, 252, 1019, 4090, 16377, 65528, 262135, 1048566, 4194293 OEIS: \underline{A024037} [1] 4ˆ n-n.

AKUX \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}3 2 \\ 4 1 \end{matrix}], [\begin{matrix}1 3 \\ 4 2 \end{matrix}]\} See also CDEF

Seq= 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 OEIS: \underline{A000523} [213] Log 2(n) rounded down.

AKWY \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}], [\begin{matrix}2 1 \\ 4 3 \end{matrix}]\} See also BDQW

Seq= 4, 16, 92, 604, 4214, 30538, 227476, 1730788, 13393690, 105089230, 834086422 Var= 3, 15, 91, 603, 4213, 30537, 227475, 1730787, 13393689, 105089229, 834086421 OEIS: \underline{A099251} [1] Bisection of A005043.

APRV \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\} See also CEFW

Seq= 4, 13, 26, 43, 64, 89, 118, 151, 188, 229, 274 OEIS: \underline{A091823} [1] a(n) = 2*nˆ 2 + 3*n - 1.

APSX \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}], [\begin{matrix}1 3 \\ 4 2 \end{matrix}]\} See also DFQW

Seq= 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304 OEIS: \underline{A000302} [2] Powers of 4.

ARSV \{[\begin{matrix}4 3 \\ 1 2 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}2 1 \\ 3 4 \end{matrix}], [\begin{matrix}2 3 \\ 4 1 \end{matrix}]\}

Seq= 4, 14, 60, 250, 1016, 4086, 16372, 65522, 262128, 1048558, 4194284 Var= 2, 12, 58, 248, 1014, 4084, 16370, 65520, 262126, 1048556, 4194282 OEIS: \underline{A100103} [1] 2ˆ (2*n)-(2*n).

BCEG \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 3 \\ 2 1 \end{matrix}]\} See also BEGJ, BEGY, BEJV, BEVY, BJTV, EJRV Seq= 4, 24, 272, 4960, 132672, 4893056, 237969664, 14756156928, 1136284574720 Var= 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905, 51943281731 OEIS: \underline{A110501} [2] Unsigned Genocchi numbers (of first kind) of even index.

BCEK \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\}

Seq= 4, 14, 62, 324, 1936, 12962, 95786, 772196, 6729124, 62920648, 627487330 Var= 2, 7, 31, 162, 968, 6481, 47893, 386098, 3364562, 31460324, 313743665 OEIS: \underline{A125275} [1] Eigensequence of triangle A039599: a(n) = Sum \{k=0..n-1\} A039599(n-1,k)*a(k) for n>0 with a(0)=1.

BDEF \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}]\} See also BDEM, BDFT, BEFK, BEKM, BFKT, DEFR, DEMR, EFKR Seq= 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024 OEIS: \underline{A068875} [2] Expansion of (1+x*C)*C, where C = (1-(1-4*x)ˆ (1/2))/(2*x) is g.f. for Catalan numbers, A000108.

BDEK \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\}

Seq= 4, 12, 38, 126, 430, 1498, 5300, 18980, 68636, 250208, 918304 Var= 2, 6, 19, 63, 215, 749, 2650, 9490, 34318, 125104, 459152 OEIS: \underline{A109262} [1] A Catalan transform of the Fibonacci numbers.

BDFK \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\}

Seq= 4, 11, 32, 99, 318, 1051, 3550, 12200, 42520, 149930, 533890 OEIS: \underline{A135339} [1] Number of Dyck paths of semilength n having no DUDU’s starting at level 1.

BEFM \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 8, 18, 46, 130, 394, 1252, 4112, 13836, 47428, 165000 Var= 2, 4, 9, 23, 65, 197, 626, 2056, 6918, 23714, 82500 OEIS: \underline{A014137} [1] Partial sums of Catalan numbers (A000108).

BEFT \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

Seq= 4, 10, 30, 98, 336, 1188, 4290, 15730, 58344, 218348, 823004 Var= 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786 OEIS: \underline{A000108} [5] Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

BERT \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 3 2 \end{matrix}], [\begin{matrix}1 2 \\ 3 4 \end{matrix}]\}

Seq= 4, 16, 80, 448, 2688, 16896, 109824, 732160, 4978688, 34398208, 240787456 OEIS: \underline{A025225} [2] a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n ≥ 2. Also a(n) = (2ˆ n)*C(n-1), where C = A000108 (Catalan numbers).

BFKM \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 9, 22, 58, 163, 483, 1494, 4783, 15740, 52956, 181391 OEIS: \underline{A059019} [1] Number of Dyck paths of semilength n with no peak at height 3.

BFLM \{[\begin{matrix}3 4 \\ 1 2 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 8, 20, 60, 200, 704, 2552, 9416, 35156, 132396, 501908 Var= 1, 2, 5, 15, 50, 176, 638, 2354, 8789, 33099, 125477 OEIS: \underline{A024718} [1] (1/2)*(1 + sum of C(2k,k)) for k = 0,1,2,...,n.

CDEK \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}]\} See also CDFK

Seq= 4, 10, 32, 106, 412, 1634, 7240, 32722, 160436, 803002, 4279024 Var= 2, 5, 16, 53, 206, 817, 3620, 16361, 80218, 401501, 2139512 OEIS: \underline{A081126} [1] Binomial transform of n!/floor(n/2)!.

CDFW \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}2 3 \\ 1 4 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\} See also CDQW, DEFW, DEQW

Seq= 4, 13, 49, 193, 769, 3073, 12289, 49153, 196609, 786433, 3145729 OEIS: \underline{A140660} [1] 3*4ˆ n+1.

CDJM \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 12, 30, 62, 112, 184, 282, 410, 572, 772, 1014 Var= 2, 6, 15, 31, 56, 92, 141, 205, 286, 386, 507 OEIS: \underline{A056520} [1] (n+2)*(2*nˆ 2-n+3)/6

CDKM \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}1 4 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 12, 40, 152, 624, 2768, 13024, 64800, 337984, 1842368, 10444416 Var= 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208 OEIS: \underline{A000898} [1] a(n) = 2(a(n-1) + (n-1)a(n-2)).

CDLM \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 1 \\ 2 4 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\}

Seq= 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244 OEIS: \underline{A005893} [1] Number of points on surface of tetrahedron: 2nˆ 2 + 2 (coordination sequence for sodalite net) for n>0.

CMPW \{[\begin{matrix}4 2 \\ 1 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}], [\begin{matrix}2 4 \\ 3 1 \end{matrix}], [\begin{matrix}3 1 \\ 4 2 \end{matrix}]\}

Seq= 4, 36, 568, 14560, 546492, 28289184, 1930982576, 168054225408, 18162775533620 Var= 2, 18, 284, 7280, 273246, 14144592, 965491288, 84027112704, 9081387766810 OEIS: \underline{A131455} [1] Number of inequivalent properly oriented and labeled planar chord diagrams whose associated planar tree is a path on n+1 ver...

DEJM \{[\begin{matrix}2 4 \\ 1 3 \end{matrix}], [\begin{matrix}3 2 \\ 1 4 \end{matrix}], [\begin{matrix}4 1 \\ 2 3 \end{matrix}], [\begin{matrix}1 3 \\ 2 4 \end{matrix}]\} See also DFJM

Seq= 4, 10, 24, 50, 92, 154, 240, 354, 500, 682, 904 Var= 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452 OEIS: \underline{A116731} [1] Number of permutations of length n which avoid the patterns 321, 2143, 3124; or avoid the patterns 132, 2314, 4312, etc.

7. Concluding remarks

Due to the computer performance limitation and the complexity of the structure of the standard puzzle, we only generate the puzzle sequences up to order |P| = 4. The following two examples show that there may have interesting sequences hidden in the case of |P| ≥ 5.

Example 1. A support, namely, P = ABDGH, that generates the secant numbers is found recently by Jiaxi Lu and Yuanzhe Ding [7]. In fact, the sequence (|ABDGH^{n}|)_{n} is

(5, 61, 1385, 50521, 2702765, 199360981, 19391512145, …)

which is the sequence A000364 of the secant numbers, shifted by two terms.

Example 2. The sequence (|AF HKQSUZ^{n}|)_{n} is

(8, 84, 1632, 51040, 2340480, 147985824, 12338740736, …)

which is the sequence A261683 shifted by one term. This sequence counts the number of whirlpool permutations of shape 2 × n, introduced by Donald Knuth in 2020 [6].

Acknowledgement. The author would like to thank the referees for carefully reading the paper and for giving helpful comments.

References

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[2] Foata, Dominique; Han, Guo-Niu, Doubloons and new q-tangent numbers, Quarterly J. Math., 62 (2011), pp. 417–432.

[3] Foata, Dominique; Han, Guo-Niu, Doubloons and q-secant numbers, Münster J. Math., 3 (2010), pp. 89– 110.

[4] Frame, J. Sutherland; Robinson, Gilbert de Beauregard; Thrall, Robert M., The hook graphs of the symmetric groups, Canad. J. Math., 6 (1954), pp. 316–324.

[5] Graham, Ron; Zang, Nan, Enumerating split-pair arrangements, J. Combin. Theory, Ser. A, 115 (2008), p. 293–303.

[6] Knuth, Donald, Whirlpool Permutations, preprint, 2020.05.17, 4 pages.

[7] Lu, Jiaxi; Ding, Yuanzhe, A skeleton model to enumerate standard puzzle sequences. preprint, arxiv:2106.09471 (2021), 21 pages.

[8] Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences. Published electronically at

   http://oeis.org/.

[9] Stanley, Richard P., Enumerative Combinatorics, vol. 2, Cambridge University Press, 1999.

[10] Wikipedia, Catalan number, Wikipedia Foundation, revision July 9, 2021.

[11] Wikipedia, Polyomino, Wikipedia Foundation, revision July 9, 2021.

[12] Wikipedia, Permutation pattern, Wikipedia Foundation, revision July 9, 2021.

[13] Wikipedia, Alternating permutation, Wikipedia Foundation, revision July 9, 2021.

[14] Wikipedia, Young tableaux, Wikipedia Foundation, revision July 9, 2021.

[15] Wikipedia, Ferrers diagram, Wikipedia Foundation, revision July 9, 2021.

2010 Mathematics Subject Classification: Primary 05A15, 05A19, 05A05; Secondary 05B50.

Keywords: standard puzzles, integer sequences, Fibonacci numbers, tangent numbers, Catalan numbers.