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A124921 Distributes the number of permutations in the alternating group; cf. A060351.

%I A124921 #20 Feb 14 2021 13:18:26
%S A124921 1,1,1,0,1,1,1,0,1,1,3,1,1,3,1,1,1,2,5,2,5,8,5,2,2,5,8,5,2,5,2,1,1,2,
%T A124921 8,4,9,18,12,6,8,19,31,17,12,21,9,3,2,10,19,14,18,30,21,6,4,14,17,10,
%U A124921 6,6,3,0
%N A124921 Distributes the number of permutations in the alternating group; cf. A060351.
%C A124921 The symmetric group distribution of permutation descents is summarized in Table A008292; for example 1 57 302 302 57 1 sums the following A060351 values:
%C A124921 1.......5.......10.......10.......5.......1
%C A124921 .......14.......35.......35.......14.......
%C A124921 .......19.......26.......40.......19.......
%C A124921 .......14.......40.......19.......14.......
%C A124921 ........5.......61.......26.......5.......
%C A124921 ................26.......61..............
%C A124921 ................19.......40..............
%C A124921 ................40.......26..............
%C A124921 ................35.......35..............
%C A124921 ................10.......10..............
%H A124921 Alois P. Heinz, <a href="/A124921/b124921.txt">Rows n = 0..14, flattened</a>
%e A124921 The distribution is based on the frequency of descents; for example, when permuting four symbols the 12 patterns are ddd ddu dud udu dud duu udd udu dud udu uud and uuu yielding the frequency distribution 1 1 3 1 1 3 1 1.
%e A124921 Triangle T(n,k) begins:
%e A124921   1;
%e A124921   1;
%e A124921   1, 0;
%e A124921   1, 1, 1, 0;
%e A124921   1, 1, 3, 1, 1, 3, 1, 1;
%e A124921   1, 2, 5, 2, 5, 8, 5, 2, 2, 5, 8, 5, 2, 5, 2, 1;
%e A124921   ...
%p A124921 b:= proc(u, o, t, h) option remember; expand(`if`(u+o=0, h,
%p A124921       add(b(u-j, o+j-1, t+1, irem(h+u-j, 2))*x^floor(2^(t-1)), j=1..u)+
%p A124921       add(b(u+j-1, o-j, t+1, irem(h+u+j-1, 2)), j=1..o)))
%p A124921     end:
%p A124921 T:= n-> (p-> seq(coeff(p, x, i), i=0..ceil(2^(n-1)-1)))(b(n, 0$2, 1)):
%p A124921 seq(T(n), n=0..7);  # _Alois P. Heinz_, Sep 09 2020
%t A124921 b[u_, o_, t_, h_] := b[u, o, t, h] = Expand[If[u+o == 0, h,
%t A124921      Sum[b[u-j, o+j-1, t+1, Mod[h+u-j, 2]]*x^Floor[2^(t-1)], {j, 1, u}]+
%t A124921      Sum[b[u+j-1, o-j, t+1, Mod[h+u+j-1, 2]], {j, 1, o}]]];
%t A124921 T[n_] := With[{p = b[n, 0, 0, 1]}, Table[Coefficient[p, x, i],
%t A124921      {i, 0, Ceiling[2^(n-1)-1]}]];
%t A124921 T /@ Range[0, 7] // Flatten (* _Jean-François Alcover_, Feb 14 2021, after _Alois P. Heinz_ *)
%Y A124921 Cf. A001710 (row sums), A008292, A060351, A011782 (row lengths).
%K A124921 tabf,look,nonn
%O A124921 0,11
%A A124921 _Alford Arnold_, Nov 16 2006
%E A124921 More terms from _Alois P. Heinz_, Sep 09 2020