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 A124921 Distributes the number of permutations in the alternating group; cf. A060351. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 11 07:51 EDT 2024. Contains 375059 sequences. (Running on oeis4.)