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

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, 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, 6, 6, 3, 0

Offset

0,11

Comments

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:

1.......5.......10.......10.......5.......1

.......14.......35.......35.......14.......

.......19.......26.......40.......19.......

.......14.......40.......19.......14.......

........5.......61.......26.......5.......

................26.......61..............

................19.......40..............

................40.......26..............

................35.......35..............

................10.......10..............

Links

Alois P. Heinz, Rows n = 0..14, flattened

Example

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.
Triangle T(n,k) begins:
  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;
  ...

Maple

b:= proc(u, o, t, h) option remember; expand(`if`(u+o=0, h,
      add(b(u-j, o+j-1, t+1, irem(h+u-j, 2))*x^floor(2^(t-1)), j=1..u)+
      add(b(u+j-1, o-j, t+1, irem(h+u+j-1, 2)), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..ceil(2^(n-1)-1)))(b(n, 0$2, 1)):
seq(T(n), n=0..7);  # Alois P. Heinz, Sep 09 2020

Mathematica

b[u_, o_, t_, h_] := b[u, o, t, h] = Expand[If[u+o == 0, h,
     Sum[b[u-j, o+j-1, t+1, Mod[h+u-j, 2]]*x^Floor[2^(t-1)], {j, 1, u}]+
     Sum[b[u+j-1, o-j, t+1, Mod[h+u+j-1, 2]], {j, 1, o}]]];
T[n_] := With[{p = b[n, 0, 0, 1]}, Table[Coefficient[p, x, i],
     {i, 0, Ceiling[2^(n-1)-1]}]];
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, Feb 14 2021, after Alois P. Heinz *)

Crossrefs

Cf. A001710 (row sums), A008292, A060351, A011782 (row lengths).

Sequence in context: A110268 A058965 A226306 * A090623 A098094 A367824

Adjacent sequences: A124918 A124919 A124920 * A124922 A124923 A124924

Keyword

tabf,look,nonn

Author

Alford Arnold, Nov 16 2006

Extensions

More terms from Alois P. Heinz, Sep 09 2020

Status

approved