A128612 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an even number of inversions.
1, 0, 1, 0, 2, 1, 1, 5, 5, 1, 1, 14, 30, 14, 1, 0, 28, 155, 147, 29, 1, 0, 56, 605, 1208, 586, 64, 1, 1, 127, 2133, 7819, 7819, 2133, 127, 1, 1, 262, 7288, 44074, 78190, 44074, 7288, 262, 1, 0, 496, 23947, 227623, 655039, 655315, 227569, 23893, 517, 1, 0, 992, 76305, 1102068, 4868556, 7862124, 4869558, 1101420, 76332, 1044, 1
Offset: 1
Examples
Triangle starts: 1; 0, 1; 0, 2, 1; 1, 5, 5, 1; 1, 14, 30, 14, 1; 0, 28, 155, 147, 29, 1; 0, 56, 605, 1208, 586, 64, 1; 1, 127, 2133, 7819, 7819, 2133, 127, 1; ...
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
Programs
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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: A128612 := 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,",A128612(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, 1-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
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Mathematica
b[u_, o_, i_] := b[u, o, i] = Expand[If[u + o == 0, 1 - 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 *)
Extensions
More terms from R. J. Mathar, Nov 01 2007