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A317166
Number of permutations of [n] with distinct lengths of increasing runs.
3
1, 1, 1, 5, 7, 27, 241, 505, 1975, 10241, 188743, 460545, 2323679, 10836141, 85023209, 2734858573, 8010483015, 45714797671, 243112435345, 1632532938001, 15831051353773, 892173483721887, 2978105991739613, 19855526019022967, 113487352591708591
OFFSET
0,4
LINKS
FORMULA
a(A000217(n)) = A317165(n).
MAPLE
g:= (n, s)-> `if`(n in s, 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
`if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
, j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..24);
MATHEMATICA
g[n_, s_] := If[MemberQ[s, n], 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
{j, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 23 2018
STATUS
approved