A249560 Number of length n permutations avoiding (231,{1},{}) and (132,{},{2}).

1, 1, 2, 4, 9, 22, 58, 163, 485, 1519, 4985, 17077, 60871, 225152, 862150, 3410641, 13913800, 58440010, 252348913, 1118802690, 5086910935, 23693925911, 112947299251, 550527774738, 2741489275969, 13936841789100, 72277551806634, 382134348251357, 2058420014680378

Offset: 0

Author

Christian Bean, Nov 01 2014

Keywords

• nonn

Comments

(231,{1},{}) is a vincular pattern. It has underlying classical pattern 231 and the extra requirement that the 2 and the 3 are adjacent in the permutation.

(132,{},{2}) is a co-vincular pattern. It has underlying classical pattern 132 and the extra requirement that the 2 and 3 are exactly one apart in the permutation.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Christian Bean, A Claesson, H Ulfarsson, Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3, arXiv preprint arXiv:1512.03226, 2015

Crossrefs

Cf. A249561, A249562, A249563.

Formula

if i = 1: a(n,k,i) = sum( a(n-1,k,j) for j in [1..k] )

if i = k: a(n,k,i) = sum( a(n-1,k-1,j) + a(n-1,k,j) for j in [1..k-1] )

otherwise: a(n,k,i) = sum( a(n-1,k,j) for j in [1..i-1] )

where n is the length, k is the number of right to left minima and i is the position of the maximum in relation to the right to left minima.

Initial Conditions: if k > n or i > k then a(n,k,i) = 0, if k = 1 then a(n,k,i) = 1.

Then a(n) = sum( sum( a(n,k,i) for i in [1..k]) for k in [1..n] ).

G.f: 1 + x * sum(x^n * F_n(1+x) for n >= 0) where F_n(q) = sum( [n,m] for m in [0..n] ). Note [n,m] is the q-binomial. - Christian Bean, Jun 03 2015

Extensions

More terms from Alois P. Heinz, Nov 01 2014