A047890 Number of permutations in S_n with longest increasing subsequence of length <= 5.
1, 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, 657723056000, 8636422912277, 117241501095189, 1639974912709122, 23570308719710838, 347217077020664880, 5231433025400049936, 80466744544235325387
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..735
- 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, 2015.
- Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
- Permutation Pattern Avoidance Library (PermPAL), 012345
- 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.
Programs
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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=0..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=0..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-François Alcover, Mar 10 2014, after Alois P. Heinz *)
Formula
a(n) ~ 9 * 5^(2*n + 25/2) / (512 * n^12 * Pi^2). - Vaclav Kotesovec, Sep 10 2014
Extensions
More terms from Naohiro Nomoto, Mar 01 2002
More terms from Alois P. Heinz, Apr 10 2012