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A321313
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Number of permutations of [n] with equal lengths of the longest increasing subsequence and the longest decreasing subsequence.
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4
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1, 0, 4, 4, 36, 256, 1282, 9864, 99976, 970528, 9702848, 113092200, 1500063930, 20985500212, 305177475748, 4733232671056, 79461918315024, 1427464201289584, 26955955609799728, 531536672155429792, 10980840178654738496, 238597651836121062824, 5446220581860028853936
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OFFSET
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1,3
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LINKS
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FORMULA
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MAPLE
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h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> `if`(l[1]=nops(l), h(l)^2, 0):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);
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MATHEMATICA
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h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
f[l_] := If[l[[1]] == Length[l], h[l]^2, 0];
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
a[n_] := g[n, n, {}];
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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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