A332471 Numbers a(n) which count the linear extensions of the zigzag poset Z of length 2n where each minimal element in Z additionally covers a new element.

1, 16, 1036, 174664, 60849880, 38013766336, 38705790148480, 59974794813746176, 134300248452273030400, 417431293378855977894400, 1743578543837236847348608000, 9530635895810801293384327628800, 66681092396823971746429574881638400, 586039304763336550135184327900472524800

Offset

1,2

Comments

The poset corresponding to a(n) is defined by the following cover relations on elements {1,2,...,3n}: {3i-2 < 3i-1 : i = 1...n} and {3i-1 < 3i : i = 1...n} and {3i > 3i+2 : i = 1...n-1}.

This sequence is the first instance of a generalization of Euler Numbers defined in the Garver et al. reference. In general, C_p(n) is the number of linear extensions of the zigzag of 2n elements, where each minimal element additionally covers a path of length p. We have a(n) = C_1(n)

References

R. P. Stanley, Enumerative combinatorics, 2nd ed., Vol. 1, Cambridge University Press, 2012.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..100

Alexander Garver, Stefan Grosser, Jacob Matherne and Alejandro Morales, Counting Linear Extensions of Posets with Determinants of Hook Lengths, arXiv:2001.08822 [math.CO], 2020.

GaYee Park, Naruse hook formula for linear extensions of mobile posets, arXiv:2104.11166 [math.CO], 2021.

Wikipedia, Alternating Permutation

Formula

a(n) = (3n)! * det(c_{i,j}) with 1<= i,j <= n, where c_{i,j} is the following matrix: for j >= i-1, c_{i,j} = Prod_{r=1...j-i+1} 1/(3r(3r-1)); otherwise c_{i,j} = 0. (Proved)

a(n) ~ (3*n)! * c * d^n, where d = 0.12759419022704514910843029597508608447727573534180502377044888068225153... and c = 1.3759920635975758250596557376465522105456574791399781031599409800963... - Vaclav Kotesovec, Feb 26 2020

Example

a(2) = 6! * det({{1/(3!), 1/(6*5*3*2)},{1, 1/(3!)}}) = 16.

Maple

a:=(k)->(3*k)!*LinearAlgebra:-Determinant(Matrix(k, k, (i, j)->`if`(j>=i-1, mul(1/(3*r*(3*r-1)), r=1..j-i+1), 0)));
seq(a(k), k=1..10);

Mathematica

nmax = 15; Table[(3*n)!*Det[Table[If[j>=i-1, Product[1/(3*r*(3*r-1)), {r, 1, j-i+1}], 0], {i, 1, n}, {j, 1, n}]], {n, 1, nmax}] (* Vaclav Kotesovec, Feb 26 2020 *)

Prog

(PARI) a(n) = (3*n)!*matdet(matrix(n, n, i, j, if (j>=i-1, prod(r=1, j-i+1, 1/(3*r*(3*r-1)))))); \\ Michel Marcus, Feb 20 2020

Crossrefs

Removing the added elements to the zigzag, this sequence would match the Euler numbers A000111.

Sequence in context: A013735 A180376 A162008 * A195619 A348862 A029701

Adjacent sequences: A332468 A332469 A332470 * A332472 A332473 A332474

Keyword

nonn,easy

Author

Stefan Grosser, Feb 13 2020

Status

approved