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

1,5

Links

Alois P. Heinz, Rows n = 1..141, flattened

Jason Fulman, Gene B. Kim, Sangchul Lee, 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.

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<=k<n. - R. J. Mathar, Nov 01 2007

Example

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;

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 *)

Crossrefs

Cf. A008292, A049061, A128612.

Sequence in context: A112964 A368845 A350791 * A231063 A295216 A230250

Adjacent sequences: A128610 A128611 A128612 * A128614 A128615 A128616

Keyword

nonn,tabl

Author

Ralf Stephan, May 08 2007

Extensions

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

Status

approved