A229001 Total sum A(n,k) of the k-th powers of lengths of ascending runs in all permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 3, 0, 1, 4, 12, 0, 1, 6, 18, 60, 0, 1, 10, 32, 96, 360, 0, 1, 18, 66, 186, 600, 2520, 0, 1, 34, 152, 426, 1222, 4320, 20160, 0, 1, 66, 378, 1110, 2964, 9086, 35280, 181440, 0, 1, 130, 992, 3186, 8254, 22818, 75882, 322560, 1814400

Offset: 0

Author

Alois P. Heinz, Sep 10 2013

Keywords

• nonn
• tabl

Examples

			A(3,2) = 32 = 9+5+5+5+5+3 = 3^2+4*(2^2+1^2)+3*1^2: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
Square array A(n,k) begins:
:    0,    0,    0,     0,     0,      0,      0, ...
:    1,    1,    1,     1,     1,      1,      1, ...
:    3,    4,    6,    10,    18,     34,     66, ...
:   12,   18,   32,    66,   152,    378,    992, ...
:   60,   96,  186,   426,  1110,   3186,   9846, ...
:  360,  600, 1222,  2964,  8254,  25620,  86782, ...
: 2520, 4320, 9086, 22818, 66050, 214410, 765506, ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A001710(n+1) for n>0, A001563, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.

Rows n=0-2 give: A000004, A000012, A052548.

Main diagonal gives: A229002.

Programs

Maple

A:= (n, k)-> add(`if`(n=t, 1, n!/(t+1)!*(t*(n-t+1)+1
             -((t+1)*(n-t)+1)/(t+2)))*t^k, t=1..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := Sum[If[n == t, 1, n!/(t + 1)!*(t*(n - t + 1) + 1 - ((t + 1)*(n - t) + 1)/(t + 2))]* t^k, {t, 1, n}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Formula

A(n,k) = Sum_{t=1..n} t^k * A122843(n,t).

For fixed k, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. - Vaclav Kotesovec, Sep 12 2013