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

%I A128613 #25 Mar 21 2025 09:12:45
%S A128613 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,
%T A128613 56,0,0,120,2160,7800,7800,2160,120,0,0,240,7320,44160,78000,44160,
%U A128613 7320,240,0,1,517,23893,227569,655315,655039,227623,23947,496,0,1,1044,76332,1101420,4869558,7862124,4868556,1102068,76305,992,0
%N A128613 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an odd number of inversions.
%H A128613 Alois P. Heinz, <a href="/A128613/b128613.txt">Rows n = 1..141, flattened</a>
%H A128613 Jason Fulman, Gene B. Kim, Sangchul Lee, and T. Kyle Petersen, <a href="https://arxiv.org/abs/1910.04258">On the joint distribution of descents and signs of permutations</a>, arXiv:1910.04258 [math.CO], 2019.
%H A128613 S. Tanimoto, <a href="http://arXiv.org/abs/math.CO/0602263">A new approach to signed Eulerian numbers</a>, arXiv:math/0602263 [math.CO], 2006.
%F A128613 a(n) = 1/2 * (A008292(n,k) - A049061(n,k)).
%F A128613 T(n,k) = 1/2 * (A008292(n,n-k) - A049061(n,n-k)), n>=1, 0<=k<n. - _R. J. Mathar_, Nov 01 2007
%e A128613 Triangle starts:
%e A128613   0;
%e A128613   1,   0;
%e A128613   1,   2,    0;
%e A128613   0,   6,    6,    0;
%e A128613   0,  12,   36,   12,    0;
%e A128613   1,  29,  147,  155,   28,    0;
%e A128613   1,  64,  586,  120,  605,   56,   0;
%e A128613   0, 120, 2160, 7800, 7800, 2160, 120, 0;
%e A128613   ...
%p A128613 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
%p A128613 # second Maple program:
%p A128613 b:= proc(u, o, i) option remember; expand(`if`(u+o=0, i,
%p A128613        add(b(u+j-1, o-j, irem(i+u+j-1, 2)), j=1..o)*x+
%p A128613        add(b(u-j, o+j-1, irem(i+u-j, 2)), j=1..u)))
%p A128613     end:
%p A128613 T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0$2)):
%p A128613 seq(T(n), n=1..14);  # _Alois P. Heinz_, May 02 2017
%t A128613 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 A128613 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0,0]];
%t A128613 Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jul 25 2017, after _Alois P. Heinz_ *)
%Y A128613 Cf. A008292, A049061, A128612.
%K A128613 nonn,tabl
%O A128613 1,5
%A A128613 _Ralf Stephan_, May 08 2007
%E A128613 Corrected and extended by _R. J. Mathar_, Nov 01 2007