A318408 - OEIS

A318408

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.

%I #18 Sep 07 2018 03:59:48

%S 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,

%T 3087,6796,3087,234,1,1,487,11637,48355,48355,11637,487,1,1,996,40804,

%U 291484,543030,291484,40804,996,1

%N 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.

%C Note that we assume the permutations are lexicographically ordered in a zero-indexed list from smallest to largest.

%C Recall that a descent in a permutation p of [n+1] is an index i in [n] such that p(i) > p(i+1).

%C 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.

%C 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.

%H L. Solus. <a href="https://arxiv.org/abs/1807.08223">Local h^*-polynomials of some weighted projective spaces</a>, arXiv:1807.08223 [math.CO], 2018. To appear in the Proceedings of the 2018 Summer Workshop on Lattice Polytopes at Osaka University (2018).

%e The triangle T(n,k) begins:

%e n\k| 1 2 3 4 5 6 7 8 9

%e ---+---------------------------------------------------------

%e 0 | 0

%e 1 | 0

%e 2 | 1

%e 3 | 1 1

%e 4 | 1 6 1

%e 5 | 1 19 19 1

%e 6 | 1 48 142 48 1

%e 7 | 1 109 730 730 109 1

%e 8 | 1 234 3087 6796 3087 234 1

%e 9 | 1 487 11637 48355 48355 11637 487 1

%e 10 | 1 996 40804 291484 543030 291484 40804 996 1

%o (Macaulay2)

%o R = QQ[z];

%o factoradicBox = n -> (

%o L := toList(1..(n!-1));

%o B := {};

%o for j in L do

%o if (j%6!=0 and j%6!=2 and j%6!=3 and j%6!=4) then B = append(B,j);

%o W := B / (i->z^(i-sum(1..(n-1),j->floor(i/((n-j)!+(n-1-j)!)))));

%o return sum(W);

%o );

%Y Cf. A008292.

%K nonn,tabf,more

%O 0,7

%A _Liam Solus_, Aug 26 2018