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Number of permutations of [n] whose lengths of increasing runs are triangular numbers.
8

%I #17 Apr 29 2020 05:17:03

%S 1,1,1,2,7,24,93,483,2832,17515,123226,978405,8312802,75966887,

%T 756376739,8070649675,91320842018,1099612368110,14054043139523,

%U 189320856378432,2682416347625463,39945105092501742,623240458310527252,10160826473676346731,172871969109661492526

%N Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

%H Alois P. Heinz, <a href="/A317130/b317130.txt">Table of n, a(n) for n = 0..400</a>

%e a(2) = 1: 21.

%e a(3) = 2: 123, 321.

%e a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.

%e a(5) = 24: 12543, 13542, 14532, 21354, 21453, 23541, 24531, 31254, 31452, 32145, 32451, 34521, 41253, 41352, 42135, 42351, 43125, 51243, 51342, 52134, 52341, 53124, 54123, 54321.

%p g:= n-> `if`(issqr(8*n+1), 1, 0):

%p b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),

%p `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+

%p add(b(u+j-1, o-j, t+1), j=1..o))

%p end:

%p a:= n-> b(n, 0$2):

%p seq(a(n), n=0..27);

%t g[n_] := If[IntegerQ @ Sqrt[8n+1], 1, 0];

%t b[u_, o_, t_] := b[u, o, t] = If[u+o==0, g[t], If[g[t]==1, Sum[b[u-j, o+j-1, 1], {j, 1, u}], 0] + Sum[b[u+j-1, o-j, t+1], {j, 1, o}]];

%t a[n_] := b[n, 0, 0];

%t a /@ Range[0, 27] (* _Jean-François Alcover_, Apr 29 2020, after _Alois P. Heinz_ *)

%Y Cf. A000217, A097597, A193374, A317111, A317128, A317129, A317131, A317132, A317446.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jul 21 2018