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%I #26 Nov 24 2016 19:37:26
%S 1,1,2,4,9,22,57,156,447,1335,4140,13290,44055,150494,528860,1909010,
%T 7068879,26820161,104156616,413634176,1678323602,6952182325,
%U 29378908664,126568357635,555540596435,2482857618713,11292588202568,52241502972732,245700673227652,1174263543570678,5700387781450739
%N Number of length n permutations avoiding (132,{2},{}) and (123,{},{1}).
%C (132,{2},{}) is a vincular pattern. It has underlying classical pattern 132 and the extra requirement that the 3 and the 2 are adjacent in the permutation.
%C (123,{},{1}) is a co-vincular pattern. It has underlying classical pattern 123 and the extra requirement that the 1 and 2 are exactly one apart in the permutation.
%H Christian Bean, <a href="/A249563/a249563.txt">Sage code</a>
%H Christian Bean, A. Claesson, H. Ulfarsson, <a href="http://arxiv.org/abs/1512.03226">Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3</a>, arXiv preprint arXiv:1512.03226, 2015
%F If x appears after x-1 then we say that x is a ceiling point.
%F aup(n,k,i,l) = sum( sum( adown(n-1,k,j,m) for m in [i+1..k] ) for j in [1..i] )
%F if i = 1 and l > i: adown(n,k,i,l) = aup(n-1,k,l,l) + sum( adown(n-1,k,j,l) for j in [i+1..k] ) + sum( a(n-1,k-1,j,l-1) for j in [1..k-1] )
%F if i = 1 and l <= 1: adown(n,k,i,l) = sum( adown(n-1,k,j,l) for j in [i+1..k] ) + sum( a(n-1,k-1,j,l-1) for j in [1..k-1] )
%F if i > 1 and l > i: adown = aup(n-1,k,l,l) + sum( adown(n-1,k,j,l) for j in [i+1..k] )
%F otherwise: adown(n,k,i,l) = sum( adown(n-1,k,j,l) for j in [i+1..k] )
%F a(n,k,i,l) = aup(n,k,i,l) + adown(n,k,i,l)
%F where n is the length, k is the number of left to right minima, i is the position of the maximum, l is the position of the first ceiling point where if l = 0 there is no ceiling point (notice positions are in relation the position of the left to right minima).
%F aup implies the maximum is a ceiling point and adown implies the maximum is not a ceiling point.
%F Initial Conditions: if k > n or i > k or i > l then aup(n,k,i,l) - adown(n,k,i,l) = 0, if i > l then aup(n,k,i,l) = 0, and finally aup(n,n-1,i,l) = 1.
%F a(n) = sum( sum( sum( a(n,k,j,m) for m in [1..k] ) for j in [1..k] ) for k in [1..n] )
%Y Cf. A249560, A249561, A249562.
%K nonn
%O 0,3
%A _Christian Bean_, Nov 01 2014
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