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 A321314 Number of permutations of [n] where the length of the longest increasing subsequence is larger than the length of the longest decreasing subsequence. 4
 0, 1, 1, 10, 42, 232, 1879, 15228, 131452, 1329136, 15106976, 182954700, 2363478435, 33096395494, 501248446126, 8094778608472, 138112754890488, 2487454752219208, 47344572399516136, 950682668010605104, 20055050996527350752, 442701537970743308588, 10202898078512473893032 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..80 Wikipedia, Longest increasing subsequence FORMULA a(n) = Sum_{k=1..n-1} A321316(n,k). a(n) = (n! - A321313(n))/2. a(n) = A321315(n) - A321313(n). MAPLE 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] `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); MATHEMATICA 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, {}]; Array[a, 25] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *) CROSSREFS Cf. A000142, A003316, A321313, A321315, A321316. Sequence in context: A003822 A087120 A222358 * A348095 A027149 A324512 Adjacent sequences: A321311 A321312 A321313 * A321315 A321316 A321317 KEYWORD nonn AUTHOR Alois P. Heinz, Nov 03 2018 STATUS approved

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Last modified June 23 00:47 EDT 2024. Contains 373629 sequences. (Running on oeis4.)