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%I #21 Oct 27 2023 18:28:52
%S 1,12,127,1222,11096,97140,830152,6977918,57968938,477479647,
%T 3908025133,31832823274,258341395508,2090604162540,16880171617952,
%U 136054564607870,1095059149237006,8803843758642693,70715260139217139,567591311612071157,4553028235287085366
%N The number of 321-avoiding linear extensions of the comb poset K_{3,n}^beta.
%H Alois P. Heinz, <a href="/A275941/b275941.txt">Table of n, a(n) for n = 1..1000</a> (first 39 terms from Colin Defant)
%H C. Defant, <a href="http://arxiv.org/abs/1608.03951">Poset Pattern-Avoidance Problems Posed by Yakoubov</a>, arXiv:1608.03951 [math.CO], 2016.
%H S. Yakoubov, <a href="http://arxiv.org/abs/1310.2979">Pattern Avoidance in Extensions of Comb-Like Posets</a>, arXiv preprint arXiv:1310.2979 [math.CO], 2013.
%F 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=3..(2n+1)} (F_{3,n}(k)*binomial(3n-k,n-1)).
%F limit_{n-> infinity} a(n)^(1/n) = 8.
%e The a(2)=12 321-avoiding linear extensions of K_{3,2}^beta are 123456, 123465, 123546, 123564, 123645, 124356, 124365, 124536, 125346, 125364, 142356, 142365
%p a:= proc(n) option remember; `if`(n<4, [0, 1, 12, 127][n+1],
%p ((n-1)*(2*n-3)*(-22080-140168*n+729723*n^4-1060811*n^3
%p +702042*n^2+28875*n^6-235565*n^5)*a(n-1) -(2*(-144000
%p -1406688*n+29671327*n^4-23732755*n^3+9830558*n^2
%p +8703775*n^6-1912636*n^7+174020*n^8-21187921*n^5))*a(n-2)
%p +(48*(3*n-7))*(2*n-1)*(2*n-5)*(3*n-8)*(385*n^4-1298*n^3
%p +1411*n^2-618*n+144)*a(n-3)) / ((2*(385*n^4-2838*n^3
%p +7615*n^2-8874*n+3856))*(n-1)*n*(2*n-3)*(2*n-1)))
%p end:
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Aug 18 2016
%t F[s_, t_, k_] :=
%t If[s <= k <= (s - 1) t + 1,
%t If[s == 2, 1,
%t Sum[F[s - 1, t, i] Sum[
%t Binomial[k - i - 1, j], {j, k - (s - 2) t - 2, t - 1}], {i,
%t s - 1, k - 1}]], 0]
%t Table[Sum[F[3, n, k] Binomial[3 n - k, n - 1], {k, 3, 2 n + 1}], {n,
%t 1, 20}]
%K nonn,easy
%O 1,2
%A _Colin Defant_, Aug 12 2016
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