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 A214015 Number of permutations A(n,k) in S_n with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 22
 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 1, 0, 1, 1, 2, 6, 14, 1, 0, 1, 1, 2, 6, 23, 42, 1, 0, 1, 1, 2, 6, 24, 103, 132, 1, 0, 1, 1, 2, 6, 24, 119, 513, 429, 1, 0, 1, 1, 2, 6, 24, 120, 694, 2761, 1430, 1, 0, 1, 1, 2, 6, 24, 120, 719, 4582, 15767, 4862, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS A(n,k) is also the sum of the squares of numbers of standard Young tableaux (SYT) of height <= k over all partitions of n. This array is a larger and reflected version of A047888. Column k>1 is asymptotic to (Product_{j=1..k} j!) * k^(2*n + k^2/2) / (Pi^((k-1)/2) * 2^((k-1)*(k+2)/2) * n^((k^2-1)/2)). - Vaclav Kotesovec, Sep 10 2014 LINKS Alois P. Heinz, Antidiagonals n = 0..70, flattened Wikipedia, Longest increasing subsequence problem Wikipedia, Young tableau EXAMPLE A(4,2) = 14 because 14 permutations of {1,2,3,4} do not contain an increasing subsequence of length > 2: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321. Permutation 1423 is not counted because it contains the noncontiguous increasing subsequence 123. A(4,2) = 14 = 2^2 + 3^2 + 1^2 because the partitions of 4 with <= 2 parts are [2,2], [3,1], [4] with 2, 3, 1 standard Young tableaux, respectively: +------+ +------+ +---------+ +---------+ +---------+ +------------+ | 1 3 | | 1 2 | | 1 3 4 | | 1 2 4 | | 1 2 3 | | 1 2 3 4 | | 2 4 | | 3 4 | | 2 .-----+ | 3 .-----+ | 4 .-----+ +------------+ +------+ +------+ +---+ +---+ +---+ Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 2, 2, 2, 2, 2, ... 0, 1, 5, 6, 6, 6, 6, 6, ... 0, 1, 14, 23, 24, 24, 24, 24, ... 0, 1, 42, 103, 119, 120, 120, 120, ... 0, 1, 132, 513, 694, 719, 720, 720, ... 0, 1, 429, 2761, 4582, 5003, 5039, 5040, ... MAPLE h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end: g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1\$n])^2, `if`(i<1, 0, add(g(n-i*j, i-1, [l[], i\$j]), j=0..n/i))): A:= (n, k)-> `if`(k>=n, n!, g(n, k, [])): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[ 1 + l[[i]] - j + Sum [If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table [Table [a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *) CROSSREFS Columns k=0-10 give: A000007, A000012, A000108, A005802, A047889, A047890, A052399, A072131, A072132, A072133, A072167. Differences between A000142 and columns k=0-9 give: A000142 (for n>0), A033312, A056986, A158005, A158432, A159139, A159175, A217675, A217676, A217677. Main diagonal and first lower diagonal give: A000142, A033312. A(2n,n-1) gives A269042(n) for n>0. Cf. A047887, A047888, A182172, A208447, A214152, A267479. Sequence in context: A320955 A288942 A294220 * A137560 A201093 A131255 Adjacent sequences: A214012 A214013 A214014 * A214016 A214017 A214018 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 01 2012 STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)