A332568 a(n) is the number of linear extensions of the zigzag poset Z of length 2n where each minimal element in Z additionally covers two new elements.

2, 220, 163800, 445021200, 3214652032800, 50918885567409600, 1554049425558455280000, 83299908055220376343200000, 7314024060095163820937236800000, 996356404501170952495143447331200000, 201612475303525750146175379983871174400000

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

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

Removing one added element per minimal element of the zigzag would result in A332471.

Sequence in context: A101393 A124188 A261936 * A274466 A307511 A293945

Adjacent sequences: A332565 A332566 A332567 * A332569 A332570 A332571

Keyword

nonn,easy

Author

Stefan Grosser, Feb 16 2020

Status

approved