A267532 Number of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length < n.

0, 0, 1, 43, 1879, 102011, 7235651, 674641325, 81537026047, 12498099730471, 2375632826877259, 548818073236649129, 151476182218777630655, 49229890784448694885163, 18608906461974462064310179, 8094874797394331233877338741, 4015057931973886657462193434111

Offset: 0

Author

Alois P. Heinz, Jan 16 2016

Keywords

• nonn

Comments

Or number of words on {1,1,2,2,...,n,n} avoiding the pattern 12...n.

Examples

			a(2) = 1: 2211.
a(3) = 43: 113322, 131322, 133122, 133212, 133221, 211332, 213132, 213312, 213321, 221133, 221313, 221331, 223113, 223131, 223311, 231132, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311322, 313122, 313212, 313221, 321132, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121, 332211.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000079, A000142, A000680, A006902, A010050, A267479, A267480, A269042.

Column k=2 of A269129.

Programs

Maple

b:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1], (
      (n^3+n^2-7*n+4)*b(n-1)-2*(2*n-3)*(n-1)^3*b(n-2))/(n-2))
    end:
a:= n-> (2*n)!/(2^n)-b(n):
seq(a(n), n=0..20);

Formula

a(n) = (2*n)! * ( 1/(2^n) - Sum_{k=0..n} (-1)^k * C(n,k) / (n+k)! ).

a(n) = A000680(n) - A006902(n).

a(n) = A267479(n,n-1) for n>0.

a(n) = Sum_{k=0..n-1} A267480(n,k).