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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 A318408 #18 Sep 07 2018 03:59:48
%S A318408 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 A318408 3087,6796,3087,234,1,1,487,11637,48355,48355,11637,487,1,1,996,40804,
%U A318408 291484,543030,291484,40804,996,1
%N 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.
%C A318408 Note that we assume the permutations are lexicographically ordered in a zero-indexed list from smallest to largest.
%C A318408 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 A318408 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 A318408 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 A318408 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 A318408 The triangle T(n,k) begins:
%e A318408   n\k|  1     2     3       4       5       6     7     8    9
%e A318408   ---+---------------------------------------------------------
%e A318408   0  |  0
%e A318408   1  |  0
%e A318408   2  |  1
%e A318408   3  |  1     1
%e A318408   4  |  1     6     1
%e A318408   5  |  1    19    19       1
%e A318408   6  |  1    48   142      48       1
%e A318408   7  |  1   109   730     730     109       1
%e A318408   8  |  1   234  3087    6796    3087     234     1
%e A318408   9  |  1   487 11637   48355   48355   11637   487     1
%e A318408   10 |  1   996 40804  291484  543030  291484 40804   996    1
%o A318408 (Macaulay2)
%o A318408 R = QQ[z];
%o A318408 factoradicBox = n -> (
%o A318408 L := toList(1..(n!-1));
%o A318408 B := {};
%o A318408 for j in L do
%o A318408 if (j%6!=0 and j%6!=2 and j%6!=3 and j%6!=4) then B = append(B,j);
%o A318408 W := B / (i->z^(i-sum(1..(n-1),j->floor(i/((n-j)!+(n-1-j)!)))));
%o A318408 return sum(W);
%o A318408 );
%Y A318408 Cf. A008292.
%K A318408 nonn,tabf,more
%O A318408 0,7
%A A318408 _Liam Solus_, Aug 26 2018