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A047890 Number of permutations in S_n with longest increasing subsequence of length <= 5. 10
1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, 657723056000, 8636422912277, 117241501095189, 1639974912709122, 23570308719710838, 347217077020664880, 5231433025400049936, 80466744544235325387 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..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, 2015.

Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

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) ~ 9 * 5^(2*n + 25/2) / (512 * n^12 * Pi^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)

      `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)))

    end:

a:= n-> g(n, 5, []):

seq(a(n), n=1..30);  # Alois P. Heinz, Apr 10 2012

# second Maple program

a:= proc(n) option remember; `if`(n<6, n!, ((-375+400*n+843*n^2

       +322*n^3+35*n^4)*a(n-1) +225*(n-1)^2*(n-2)^2*a(n-3)

       -(259*n^2+622*n+45)*(n-1)^2*a(n-2))/ ((n+6)^2*(n+4)^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, 5], {n, 1, 30}] (* Jean-Fran├žois Alcover, Mar 10 2014, after Alois P. Heinz *)

CROSSREFS

A column of A047888. Cf. A005802, A052399.

Column k=5 of A214015.

Sequence in context: A224287 A248838 A052398 * A071088 A177533 A122417

Adjacent sequences:  A047887 A047888 A047889 * A047891 A047892 A047893

KEYWORD

nonn,easy

AUTHOR

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane.

EXTENSIONS

More terms from Naohiro Nomoto, Mar 01 2002

More terms from Alois P. Heinz, Apr 10 2012

STATUS

approved

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Last modified December 14 07:38 EST 2018. Contains 318090 sequences. (Running on oeis4.)