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.

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

Offset: 0

Author

Liam Solus, Aug 26 2018

Keywords

• nonn
• tabf
• more

Comments

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.

Examples

			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
		

Links

• L. Solus. Local h^*-polynomials of some weighted projective spaces, arXiv:1807.08223 [math.CO], 2018. To appear in the Proceedings of the 2018 Summer Workshop on Lattice Polytopes at Osaka University (2018).

Crossrefs

Cf. A008292.

Programs

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);
);