A052399 Number of permutations in S_n with longest increasing subsequence of length <= 6.

1, 1, 2, 6, 24, 120, 720, 5039, 40270, 361302, 3587916, 38957991, 457647966, 5763075506, 77182248916, 1091842643475, 16219884281650, 251774983140578, 4066273930979460, 68077194367392864, 1177729684507324152, 20995515989327134152, 384762410996641402384

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.

Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.

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.

Index entries for sequences related to Young tableaux.

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 from Vaclav Kotesovec, Sep 10 2014

Status

approved