A130477 T(n,k) is the number of permutations of [n] with maximum descent k, T(n,k) for n >= 0 and 0 <= k <= n, triangle read by rows.

1, 1, 1, 1, 2, 3, 1, 3, 8, 12, 1, 4, 15, 40, 60, 1, 5, 24, 90, 240, 360, 1, 6, 35, 168, 630, 1680, 2520, 1, 7, 48, 280, 1344, 5040, 13440, 20160, 1, 8, 63, 432, 2520, 12096, 45360, 120960, 181440, 1, 9, 80, 630, 4320, 25200, 120960, 453600, 1209600, 1814400, 1, 10, 99, 880, 6930, 47520, 277200, 1330560, 4989600, 13305600, 19958400

Offset: 0

Author

Gary W. Adamson, May 28 2007

Keywords

• nonn
• tabl

Comments

Previous name was: Triangle generated from finite differences of A130461.

Decomposition of the permutations.

From Liam Solus, Aug 23 2018: (Start)

For k>0, T(n,k) equals the number of permutations p of [n] such that the largest index i for which p(i)>p(i+1) is k; i.e., T(n,k) is the number of permutations of [n] with maximum descent being k. See Lemma 3.4 of the paper by L. Solus below.

When T(n,k) is taken as the weight of coordinate x_k for k = 0,...,n-1 in an (n-1)-dimensional weighted projective space, the result is the toric variety defined by an n-dimensional simplex whose Ehrhart h^*-polynomial is the n-th Eulerian polynomial. See Theorem 3.5 of the paper by L. Solus below.

(End)

Examples

			First few rows of the triangle A130461 = (1; 1, 1; 1, 1, 1; 1, 1, 2, 1; 1, 1, 2, 3, 1; 1, 1, 2, 6, 4, 1;...). Deleting the left border and taking finite differences at the top of each remaining column, we get the first few rows of this triangle:
1;
1, 1;
1, 2,  3;
1, 3,  8,  12;
1, 4, 15,  40,  60;
1, 5, 24,  90, 240,  360;
1, 6, 35, 168, 630, 1680, 2520;
...
		

Links

• L. Solus, Simplices for numeral systems, Transactions of the American Mathematical Society. DOI: https://doi.org/10.1090/tran/7424 (2017).

Crossrefs

The triangle with each row reversed is A092582.

Cf. A000142 (row sums), A001710 (main diagonal), A008292.

Cf. A130460, A130461, A130476, A130478.

Programs

Maple

T := (n,k) -> (n-k+1+0^k)*((n+1)!/(n-k+2)!):
seq(seq(T(n,k),k=0..n),n=0..10); # Peter Luschny, Sep 17 2018

Mathematica

Flatten[Table[Table[(n - k + 1 + 0^k)*(n + 1)!/(n - k + 2)!, {k,0,n}], {n, 0, 10}], 1] (* Olivier Gérard, Aug 04 2012 *)

PARI

{T(n, k) = if( n<1 || k>n, 0, k==1, 1, n! * (n+1-k) / (n+2-k)!)}; /* Michael Somos, Jun 25 2017 */

Formula

Each term in n-th row divides n!.

Given triangle A130461 and deleting the left border (1,1,1,...) take finite differences by columns and reorient into rows.

T(n,k) = (n-k+1+0^k)*((n+1)!/(n-k+2)!) - Olivier Gérard, Aug 04 2012

Extensions

New name using a comment by Liam Solus, Peter Luschny, Sep 17 2018