A128613 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an odd number of inversions.

0, 1, 0, 1, 2, 0, 0, 6, 6, 0, 0, 12, 36, 12, 0, 1, 29, 147, 155, 28, 0, 1, 64, 586, 1208, 605, 56, 0, 0, 120, 2160, 7800, 7800, 2160, 120, 0, 0, 240, 7320, 44160, 78000, 44160, 7320, 240, 0, 1, 517, 23893, 227569, 655315, 655039, 227623, 23947, 496, 0, 1, 1044, 76332, 1101420, 4869558, 7862124, 4868556, 1102068, 76305, 992, 0

Offset: 1

Author

Ralf Stephan, May 08 2007

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  0;
  1,   0;
  1,   2,    0;
  0,   6,    6,    0;
  0,  12,   36,   12,    0;
  1,  29,  147,  155,   28,    0;
  1,  64,  586,  120,  605,   56,   0;
  0, 120, 2160, 7800, 7800, 2160, 120, 0;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Jason Fulman, Gene B. Kim, Sangchul Lee, and T. Kyle Petersen, On the joint distribution of descents and signs of permutations, arXiv:1910.04258 [math.CO], 2019.
• S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006.

Crossrefs

Cf. A008292, A049061, A128612.

Programs

Maple

A008292 := proc(n,k) local j; add( (-1)^j*(k-j)^n*binomial(n+1,j),j=0..k) ; end: A049061 := proc(n,k) if k <= 0 or n <=0 or k > n then 0; elif n = 1 then 1 ; elif n mod 2 = 0 then A049061(n-1,k)-A049061(n-1,k-1) ; else k*A049061(n-1,k)+(n-k+1)*A049061(n-1,k-1) ; fi ; end: A128613 := proc(n,k) (A008292(n,n-k)-A049061(n,n-k))/2 ; end: for n from 1 to 11 do for k from 0 to n-1 do printf("%d,",A128613(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007
# second Maple program:
b:= proc(u, o, i) option remember; expand(`if`(u+o=0, i,
       add(b(u+j-1, o-j, irem(i+u+j-1, 2)), j=1..o)*x+
       add(b(u-j, o+j-1, irem(i+u-j, 2)), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, May 02 2017

Mathematica

b[u_, o_, i_] := b[u, o, i] = Expand[If[u + o == 0, i, Sum[b[u + j - 1, o - j, Mod[i + u + j - 1, 2]], {j, 1, o}]*x + Sum[b[u - j, o + j - 1, Mod[i + u - j, 2]], {j, 1, u}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0,0]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *)

Formula

a(n) = 1/2 * (A008292(n,k) - A049061(n,k)).

T(n,k) = 1/2 * (A008292(n,n-k) - A049061(n,n-k)), n>=1, 0<=kR. J. Mathar, Nov 01 2007

Extensions

Corrected and extended by R. J. Mathar, Nov 01 2007