OFFSET
1,1
COMMENTS
The poset corresponding to a(n) is defined by the following cover relations on elements {1,2,...,4n}: {4i-3 < 4i-1 : i = 1...n} and {4i-2 < 4i-1 : i = 1...n} and {4i-1 < 4i : i = 1...n} and {4i > 4i-1 : i = 1...n-1}.
This sequence is an instance of a generalization of Euler Numbers defined in the Garver et al. reference. In general, A_k(n) is the number of linear extensions of the zigzag of 2n elements, where each minimal element additionally covers k new elements. Specifically, a(n) = A_2(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], Jan 2020.
GaYee Park, Naruse hook formula for linear extensions of mobile posets, arXiv:2104.11166 [math.CO], 2021.
Wikipedia, Alternating Permutation
FORMULA
a(n) = (4n)! * det(c_{i,j}) with 1<= i,j <= n, where c_{i,j} is the following matrix: for j >= i-1, c_{i,j} = Product_{r=1..j-i+1} 1/(4r(4r-1)); otherwise c_{i,j} = 0. (Proved)
a(n) ~ (4*n)! * c * d^n, where d = 0.0621081230059627257075494363450193617160421717754186757880676835858048... and c = 1.42983395270155716735034676344701283104553855261001105886616... - Vaclav Kotesovec, Feb 26 2020
EXAMPLE
A_2(2) = 8! * det({{1/(4*3), 1/(8*7*4*3)},{1, 1/(4*3)}}) = 220.
MAPLE
a:=(k)->(4*k)!*LinearAlgebra:-Determinant(Matrix(k, k, (i, j)->`if`(j>=i-1, mul(1/(4*r*(4*r-1)), r=1..j-i+1), 0)));
seq(a(k), k=1..10);
MATHEMATICA
nmax = 10; Table[(4*n)!*Det[Table[If[j>=i-1, Product[1/(4*r*(4*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) = (4*n)!*matdet(matrix(n, n, i, j, if (j>=i-1, prod(r=1, j-i+1, 1/(4*r*(4*r-1)))))); \\ Michel Marcus, Feb 20 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stefan Grosser, Feb 16 2020
STATUS
approved