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A279563 Number of length n inversion sequences avoiding the patterns 102, 201, and 210.

%I A279563 #18 Jun 08 2025 22:11:54
%S A279563 1,1,2,6,22,85,328,1253,4754,17994,68158,258808,985906,3768466,
%T A279563 14451386,55585014,214377618,828795169,3211030684,12464308997,
%U A279563 48465092366,188733879657,735977084412,2873525548315,11231884145434,43947466923095,172115939825516
%N A279563 Number of length n inversion sequences avoiding the patterns 102, 201, and 210.
%C A279563 A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i<j<k) such that e_i > e_j <> e_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 102, 201, and 210.
%H A279563 Alois P. Heinz, <a href="/A279563/b279563.txt">Table of n, a(n) for n = 0..1664</a>
%H A279563 Megan A. Martinez, Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.
%F A279563 a(n) ~ 4^n / (3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 07 2021
%F A279563 G.f.: (2 - 15*x + 32*x^2 - 16*x^3 + x * (1 - 2*x) * (1 + 2*x) * (1 - 4*x)^(1/2)) / (2 * (1 - x)^2 * (1 - 2*x) * (1 - 4*x)). - _Nathan J. Britt_, Jun 08 2025
%e A279563 The length 4 inversion sequences avoiding (102, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
%p A279563 a:= proc(n) option remember; `if`(n<4, n!,
%p A279563       ((2*(12*n^3-91*n^2+213*n-149))*a(n-1)
%p A279563       -(3*(21*n^3-162*n^2+392*n-291))*a(n-2)
%p A279563       +(2*(33*n^3-257*n^2+633*n-484))*a(n-3)
%p A279563       -(4*(2*n-7))*(3*n^2-13*n+13)*a(n-4))
%p A279563        / ((n-1)*(3*n^2-19*n+29)))
%p A279563     end:
%p A279563 seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 22 2017
%t A279563 a[n_] := a[n] = If[n < 4, n!, ((2*(12*n^3 - 91*n^2 + 213*n - 149))*a[n-1] - (3*(21*n^3 - 162*n^2 + 392*n - 291))*a[n-2] + (2*(33*n^3 - 257*n^2 + 633*n - 484))*a[n-3] - (4*(2*n - 7))*(3*n^2 - 13*n + 13)*a[n-4]) / ((n - 1)*(3*n^2 - 19*n + 29))]; Array[a, 30, 0] (* _Jean-François Alcover_, Nov 06 2017, after _Alois P. Heinz_ *)
%Y A279563 Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.
%K A279563 nonn
%O A279563 0,3
%A A279563 _Megan A. Martinez_, Feb 09 2017
%E A279563 a(10)-a(26) from _Alois P. Heinz_, Feb 22 2017