OFFSET
0,3
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
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian Bean, Nov 01 2014
EXTENSIONS
More terms from Alois P. Heinz, Nov 01 2014
STATUS
approved