OFFSET
1,2
LINKS
C. Defant, Poset Pattern-Avoidance Problems Posed by Yakoubov, arXiv:1608.03951 [math.CO], 2016.
S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv preprint arXiv:1310.2979 [math.CO], 2013.
FORMULA
Define F_{2,t}(k)=1 if 2<=k<=t+1 and 0 otherwise. For s>=3, let F_{s,t}(k)=Sum_{i=(s-1)..(k-1)}(F_{s-1,t}(i)*Sum_{j=(k-(s-2)t-2)..(t-1)}(Binomial(k-i-1,j))). Then a(n)=Sum_{k=4..(3n+1)}(F_{4,n}(k)*Binomial(4n-k,n-1)).
lim_{n->inf}(a(n)^(1/n))=16.
EXAMPLE
One of the a(2)=55 321-avoiding linear extensions of K_{4,2}^beta is 12534678 because this permutation avoids the pattern 321, the entries 1,2,3,4 appear in increasing order, 1 precedes 5, 2 precedes 6, 3 precedes 7, and 4 precedes 8.
MATHEMATICA
F[s_, t_, k_] :=
If[s <= k <= (s - 1) t + 1,
If[s == 2, 1,
Sum[F[s - 1, t, i] Sum[
Binomial[k - i - 1, j], {j, k - (s - 2) t - 2, t - 1}], {i,
s - 1, k - 1}]], 0]
Table[Sum[F[4, n, k] Binomial[4 n - k, n - 1], {k, 4, 3 n + 1}], {n, 1, 17}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Colin Defant, Aug 13 2016
STATUS
approved