A275942 The number of 321-avoiding linear extensions of the comb poset K_{4,n}^beta.

1, 55, 1866, 49523, 1147175, 24446239, 492996938, 9566197039, 180473841477, 3333072098404, 60544351368853, 1085308194335997, 19246250384730902, 338260488991568790, 5900404989342994004, 102262917165512555831, 1762556960555529202081, 30231974203021095081766, 516347665987538314322805, 8785795492453186831736382

Offset

1,2

Links

Table of n, a(n) for n=1..20.

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

Sequence in context: A004352 A163722 A321034 * A217758 A346325 A240687

Adjacent sequences: A275939 A275940 A275941 * A275943 A275944 A275945

Keyword

nonn

Author

Colin Defant, Aug 13 2016

Status

approved