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A279561
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Number of length n inversion sequences avoiding the patterns 101, 102, 201, and 210.
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24
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1, 1, 2, 6, 21, 77, 287, 1079, 4082, 15522, 59280, 227240, 873886, 3370030, 13027730, 50469890, 195892565, 761615285, 2965576715, 11563073315, 45141073925, 176423482325, 690215089745, 2702831489825, 10593202603775, 41550902139551, 163099562175851
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OFFSET
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0,3
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COMMENTS
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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. This is the same as the set of length n inversion sequences avoiding 101, 102, 201, and 210.
It is conjectured that a_n also 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 021 and 120.
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LINKS
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FORMULA
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a(n) = 1 + Sum_{i=1..n-1} binomial(2i, i-1).
G.f.: (1-4*x+sqrt(-16*x^3+20*x^2-8*x+1))/(2*(x-1)*(4*x-1)).
D-finite with recurrence: n*a(n) +(-7*n+6)*a(n-1) +2*(7*n-13)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 21 2020
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EXAMPLE
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The length 4 inversion sequences avoiding (101, 102, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
The length 4 inversion sequences avoiding (021, 120) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0022, 0023, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0122, 0123.
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MAPLE
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a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
((5*n^2-12*n+6)*a(n-1)-(4*n^2-10*n+6)*a(n-2))/((n-2)*n))
end:
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MATHEMATICA
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a[n_] := 1 + Sum[Binomial[2i, i-1], {i, 0, n-1}];
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CROSSREFS
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Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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