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A328433
Number of inversion sequences of length n avoiding the consecutive patterns 011 and 012.
16
1, 1, 2, 4, 11, 37, 157, 791, 4676, 31490, 238814, 2009074, 18585645, 187366675, 2045016693, 24018394333, 302051731428, 4049206907012, 57642586053512, 868375941780450, 13801973373609889, 230808858283551859, 4051069379668626948, 74459335679007458268
OFFSET
0,3
COMMENTS
A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i < e_{i+1} <= e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 011 and 012.
LINKS
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.
FORMULA
a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n / n^alfa, where alfa = A073016 = Sum_{k>=1} 1/binomial(2*k, k) = 1/3 + 2*Pi/3^(5/2) = 0.73639985871871507790... and c = 2.21611825460684222558745179... - Vaclav Kotesovec, Oct 19 2019
EXAMPLE
The a(4)=11 length 4 inversion sequences avoiding the consecutive patterns 011 and 012 are 0000, 0100, 0010, 0020, 0001, 0101, 0021, 0002, 0102, 0003, and 0103.
MAPLE
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
`if`(t and i < x, 0, b(n - 1, i, i <= x)), i = 0 .. n - 1))
end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);
MATHEMATICA
b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i < x, 0, b[n - 1, i, i <= x]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020, after Alois P. Heinz in A328357 *)
KEYWORD
nonn
AUTHOR
Juan S. Auli, Oct 16 2019
STATUS
approved