A332471 - OEIS

%I #33 Jul 26 2021 10:59:55

%S 1,16,1036,174664,60849880,38013766336,38705790148480,

%T 59974794813746176,134300248452273030400,417431293378855977894400,

%U 1743578543837236847348608000,9530635895810801293384327628800,66681092396823971746429574881638400,586039304763336550135184327900472524800

%N 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.

%C 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}.

%C 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)

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

%H Michael De Vlieger, <a href="/A332471/b332471.txt">Table of n, a(n) for n = 1..100</a>

%H Alexander Garver, Stefan Grosser, Jacob Matherne and Alejandro Morales, <a href="https://arxiv.org/abs/2001.08822">Counting Linear Extensions of Posets with Determinants of Hook Lengths</a>, arXiv:2001.08822 [math.CO], 2020.

%H GaYee Park, <a href="https://arxiv.org/abs/2104.11166">Naruse hook formula for linear extensions of mobile posets</a>, arXiv:2104.11166 [math.CO], 2021.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating Permutation</a>

%F 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)

%F a(n) ~ (3*n)! * c * d^n, where d = 0.12759419022704514910843029597508608447727573534180502377044888068225153... and c = 1.3759920635975758250596557376465522105456574791399781031599409800963... - _Vaclav Kotesovec_, Feb 26 2020

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

%p 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)));

%p seq(a(k),k=1..10);

%t 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 *)

%o (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

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

%K nonn,easy

%O 1,2

%A _Stefan Grosser_, Feb 13 2020