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A321314
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Number of permutations of [n] where the length of the longest increasing subsequence is larger than the length of the longest decreasing subsequence.
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4
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0, 1, 1, 10, 42, 232, 1879, 15228, 131452, 1329136, 15106976, 182954700, 2363478435, 33096395494, 501248446126, 8094778608472, 138112754890488, 2487454752219208, 47344572399516136, 950682668010605104, 20055050996527350752, 442701537970743308588, 10202898078512473893032
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n-1} A321316(n,k).
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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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