A047889 Number of permutations in S_n with longest increasing subsequence of length <= 4.

1, 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, 172188608886, 1801013405436, 19274897768196, 210573149141896, 2343553478425816, 26525044132374656, 304856947930144656

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Or, number of permutations in S_n that avoid the pattern 12345, - N. J. A. Sloane, Mar 19 2015

Also, the dimension of the space of SL(4)-invariants in V^m ⊗ (V^*)^m, where V is the standard 4-dimensional representation of SL(4) and V^* its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

Examples

			G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 119*x^5 + 694*x^6 + 4582*x^7 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..300
• 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.
• Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
• Permutation Pattern Avoidance Library (PermPAL), 12345
• 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.

Crossrefs

A column of A047888.

Cf. A005802, A047890, A052399.

Column k=4 of A214015.

Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208. - N. J. A. Sloane, Mar 19 2015

Programs

Maple

A:=rsolve({a(0) = 1, a(1) = 1, (n^3 + 16*n^2 + 85*n + 150)*a(n + 2) = (20*n^3 + 182*n^2 + 510*n + 428)*a(n + 1) - (64*n^3 + 256*n^2 + 320*n +128)*a(n)}, a(n), makeproc): # Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

Mathematica

Flatten[{1,RecurrenceTable[{64*(-1+n)^2*n*a[-2+n]-2*(-12 + 11*n + 31*n^2 + 10*n^3)*a[-1+n] + (3+n)^2*(4+n)*a[n]==0,a[1]==1,a[2]==2},a,{n,20}]}] (* Vaclav Kotesovec, Sep 10 2014 *)

PARI

{a(n) = my(v); if( n<2, n>=0, v = vector(n+1, k, 1); for(k=2, n, v[k+1] = ((20*k^3 + 62*k^2 + 22*k - 24) * v[k] - 64*k*(k-1)^2 * v[k-1]) / ((k+3)^2 * (k+4))); v[n+1])}; /* Michael Somos, Apr 19 2015 */

Formula

a(0)=1, a(1)=1, (n^3 + 16*n^2 + 85*n + 150)*a(n+2) = (20*n^3 + 182*n^2 + 510*n + 428)*a(n+1) - (64*n^3 + 256*n^2 + 320*n + 128)*a(n). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

a(n) = (64*(n+1)*(2*n^3 + 21*n^2 + 76*n + 89)*A002895(n) -(8*n^4 + 104*n^3 + 526*n^2 + 1098*n + 776) *A002895(n+1)) / (3*(n+2)^3*(n+3)^3*(n+4)). - Mark van Hoeij, Jun 02 2010

a(n) ~ 3 * 2^(4*n + 9) / (n^(15/2) * Pi^(3/2)). - Vaclav Kotesovec, Sep 10 2014

Extensions

More terms from Naohiro Nomoto, Mar 01 2002

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar