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A317130
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Number of permutations of [n] whose lengths of increasing runs are triangular numbers.
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8
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1, 1, 1, 2, 7, 24, 93, 483, 2832, 17515, 123226, 978405, 8312802, 75966887, 756376739, 8070649675, 91320842018, 1099612368110, 14054043139523, 189320856378432, 2682416347625463, 39945105092501742, 623240458310527252, 10160826473676346731, 172871969109661492526
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OFFSET
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0,4
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LINKS
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EXAMPLE
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a(2) = 1: 21.
a(3) = 2: 123, 321.
a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.
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.
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MAPLE
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g:= n-> `if`(issqr(8*n+1), 1, 0):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
`if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
add(b(u+j-1, o-j, t+1), j=1..o))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);
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MATHEMATICA
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g[n_] := If[IntegerQ @ Sqrt[8n+1], 1, 0];
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}]];
a[n_] := b[n, 0, 0];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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