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A317166 Number of permutations of [n] with distinct lengths of increasing runs.

%I A317166 #15 Sep 01 2021 16:07:56
%S A317166 1,1,1,5,7,27,241,505,1975,10241,188743,460545,2323679,10836141,
%T A317166 85023209,2734858573,8010483015,45714797671,243112435345,
%U A317166 1632532938001,15831051353773,892173483721887,2978105991739613,19855526019022967,113487352591708591
%N A317166 Number of permutations of [n] with distinct lengths of increasing runs.
%H A317166 Alois P. Heinz, <a href="/A317166/b317166.txt">Table of n, a(n) for n = 0..66</a>
%F A317166 a(A000217(n)) = A317165(n).
%p A317166 g:= (n, s)-> `if`(n in s, 0, 1):
%p A317166 b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
%p A317166       `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
%p A317166        , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
%p A317166     end:
%p A317166 a:= n-> b(n, 0$2, {}):
%p A317166 seq(a(n), n=0..24);
%t A317166 g[n_, s_] := If[MemberQ[s, n], 0, 1];
%t A317166 b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
%t A317166      If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
%t A317166      {j, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];
%t A317166 a[n_] := b[n, 0, 0, {}];
%t A317166 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Sep 01 2021, after _Alois P. Heinz_ *)
%Y A317166 Cf. A000217, A007838, A317165.
%K A317166 nonn
%O A317166 0,4
%A A317166 _Alois P. Heinz_, Jul 23 2018