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A317273 Number of permutations of [n*(n+1)/2] whose lengths of increasing runs are the positive integers from 1 to n.

%I A317273 #17 Sep 01 2021 16:07:22
%S A317273 1,1,4,202,163692,2487100956,832252747110528,7116720347983770858600,
%T A317273 1776529280247277318394451118272,
%U A317273 14580103976468323893693256154922439405632,4377460729080839690885111988468699720430287682744896,52959485251272238069446517666752040946228209263610778166878160384
%N A317273 Number of permutations of [n*(n+1)/2] whose lengths of increasing runs are the positive integers from 1 to n.
%H A317273 Alois P. Heinz, <a href="/A317273/b317273.txt">Table of n, a(n) for n = 0..14</a>
%F A317273 a(n) = A317327(A000217(n),n).
%F A317273 a(n) <= A317165(n).
%p A317273 g:= (n, s)-> `if`(n in s, 1, 0):
%p A317273 b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
%p A317273       `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s minus {t})
%p A317273        , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
%p A317273     end:
%p A317273 a:= n-> b(n*(n+1)/2, 0$2, {$0..n}):
%p A317273 seq(a(n), n=0..10);
%t A317273 g[n_, s_] := If[MemberQ[s, n], 1, 0];
%t A317273 b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
%t A317273      If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Complement~ {t}],
%t A317273      {j, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];
%t A317273 a[n_] := b[n(n+1)/2, 0, 0, Range[0, n]];
%t A317273 Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Sep 01 2021, after _Alois P. Heinz_ *)
%Y A317273 Cf. A000027, A000217, A317165.
%K A317273 nonn
%O A317273 0,3
%A A317273 _Alois P. Heinz_, Jul 25 2018