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A318408
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Triangle read by rows: T(n,k) is the number of permutations of [n+1] with index in the lexicographic ordering of permutations being congruent to 1 or 5 modulo 6 that have exactly k descents; k > 0.
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0
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0, 0, 1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 48, 142, 48, 1, 1, 109, 730, 730, 109, 1, 1, 234, 3087, 6796, 3087, 234, 1, 1, 487, 11637, 48355, 48355, 11637, 487, 1, 1, 996, 40804, 291484, 543030, 291484, 40804, 996, 1
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OFFSET
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0,7
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COMMENTS
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Note that we assume the permutations are lexicographically ordered in a zero-indexed list from smallest to largest.
Recall that a descent in a permutation p of [n+1] is an index i in [n] such that p(i) > p(i+1).
The n-th row of the triangle T(n,k) is the coefficient vector of the local h^*-polynomial (i.e., the box polynomial) of the factoradic n-simplex. Each row is known to be symmetric and unimodal. Moreover the local h^*-polynomial of the factoradic n-simplex has only real roots. See the paper by L. Solus below for definitions and proofs of these statements.
The n-th row of T(n,k) is the coefficient sequence of a restriction of the n-th Eulerian polynomial, which is given by the n-th row of A008292.
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LINKS
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EXAMPLE
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The triangle T(n,k) begins:
n\k| 1 2 3 4 5 6 7 8 9
---+---------------------------------------------------------
0 | 0
1 | 0
2 | 1
3 | 1 1
4 | 1 6 1
5 | 1 19 19 1
6 | 1 48 142 48 1
7 | 1 109 730 730 109 1
8 | 1 234 3087 6796 3087 234 1
9 | 1 487 11637 48355 48355 11637 487 1
10 | 1 996 40804 291484 543030 291484 40804 996 1
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PROG
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(Macaulay2)
R = QQ[z];
factoradicBox = n -> (
L := toList(1..(n!-1));
B := {};
for j in L do
if (j%6!=0 and j%6!=2 and j%6!=3 and j%6!=4) then B = append(B, j);
W := B / (i->z^(i-sum(1..(n-1), j->floor(i/((n-j)!+(n-1-j)!)))));
return sum(W);
);
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CROSSREFS
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KEYWORD
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nonn,tabf,more
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AUTHOR
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STATUS
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approved
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