

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.


7



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
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OFFSET

0,5


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,...,n1 in an (n1)dimensional weighted projective space, the result is the toric variety defined by an ndimensional simplex whose Ehrhart h^*polynomial is the nth Eulerian polynomial. See Theorem 3.5 of the paper by L. Solus below.
(End)


LINKS

Table of n, a(n) for n=0..65.
L. Solus, Simplices for numeral systems, Transactions of the American Mathematical Society. DOI: https://doi.org/10.1090/tran/7424 (2017).


FORMULA

Each term in nth 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) = (nk+1+0^k)*((n+1)!/(nk+2)!)  Olivier Gérard, Aug 04 2012


EXAMPLE

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;
...


MAPLE

T := (n, k) > (nk+1+0^k)*((n+1)!/(nk+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 *)


PROG

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


CROSSREFS

The triangle with each row reversed is A092582.
Cf. A000142 (row sums), A001710 (main diagonal), A008292.
Cf. A130460, A130461, A130476, A130478.
Sequence in context: A209419 A119011 A300866 * A226513 A058127 A244490
Adjacent sequences: A130474 A130475 A130476 * A130478 A130479 A130480


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, May 28 2007


EXTENSIONS

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


STATUS

approved



