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 A052399 Number of permutations in S_n with longest increasing subsequence of length <= 6. 9
 1, 1, 2, 6, 24, 120, 720, 5039, 40270, 361302, 3587916, 38957991, 457647966, 5763075506, 77182248916, 1091842643475, 16219884281650, 251774983140578, 4066273930979460, 68077194367392864, 1177729684507324152, 20995515989327134152, 384762410996641402384 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Previous name was: Related to Young tableaux of bounded height. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..250 F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7. Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015. Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016. FORMULA a(n) ~ 5 * 2^(2*n + 6) * 3^(2*n + 21) / (n^(35/2) * Pi^(5/2)). - Vaclav Kotesovec, Sep 10 2014 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:= proc(n, i, l) option remember;       `if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1\$n])^2,        g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))     end: a:= n-> g(n, 6, []): seq(a(n), n=0..25); # Alois P. Heinz, Apr 10 2012 # second Maple program a:= proc(n) option remember; `if`(n<7, n!,       ((56*n^5-9408+11032*n+19028*n^2+7360*n^3+1092*n^4)*a(n-1)        -4*(196*n^3+1608*n^2+3167*n+444)*(n-1)^2*a(n-2)        +1152*(2*n+3)*(n-1)^2*(n-2)^2*a(n-3))/ ((n+9)*(n+8)^2*(n+5)^2))     end: seq(a(n), n=1..30);  # Alois P. Heinz, Sep 26 2012 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[a[n, 6], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *) CROSSREFS Cf. A005802, A047889, A047890. Column k=6 of A214015. Sequence in context: A164873 A226438 A248839 * A177553 A090583 A248775 Adjacent sequences:  A052396 A052397 A052398 * A052400 A052401 A052402 KEYWORD nonn AUTHOR N. J. A. Sloane, Mar 13 2000 EXTENSIONS More terms from Alois P. Heinz, Apr 10 2012 New name by Vaclav Kotesovec, Sep 10 2014 STATUS approved

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Last modified November 18 05:08 EST 2017. Contains 294853 sequences.