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A332568
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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.
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1
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2, 220, 163800, 445021200, 3214652032800, 50918885567409600, 1554049425558455280000, 83299908055220376343200000, 7314024060095163820937236800000, 996356404501170952495143447331200000, 201612475303525750146175379983871174400000
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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R. P. Stanley, Enumerative combinatorics, 2nd ed., Vol. 1, Cambridge University Press, 2012.
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LINKS
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FORMULA
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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
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EXAMPLE
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A_2(2) = 8! * det({{1/(4*3), 1/(8*7*4*3)},{1, 1/(4*3)}}) = 220.
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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