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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. 2
1, 16, 1036, 174664, 60849880, 38013766336, 38705790148480, 59974794813746176, 134300248452273030400, 417431293378855977894400, 1743578543837236847348608000, 9530635895810801293384327628800, 66681092396823971746429574881638400, 586039304763336550135184327900472524800 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 17 16:12 EDT 2022. Contains 356189 sequences. (Running on oeis4.)