A279554 Number of length n inversion sequences avoiding the patterns 010, 101, 120, 201, and 210.

1, 1, 2, 5, 15, 51, 188, 733, 2979, 12495, 53708, 235396, 1048168, 4728757, 21569339, 99309057, 460932778, 2154402107, 10131719847, 47906876978, 227620982129, 1086195559709, 5203539541856, 25016401765946, 120655622545716, 583641912889094, 2830843990419690, 13764577661078075

Offset

0,3

Comments

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 010, 101, 120, 201, and 210.

Links

Nicholas R. Beaton, Table of n, a(n) for n = 0..1000

Nathan Britt and Nicholas Beaton, Completing the enumeration of inversion sequences avoiding triples of relations, arXiv:2512.21943 [math.CO], 2025. See p. 3.

Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016-2018.

Formula

G.f. F(x) is a root of the polynomial (4*x^3-2*x^2+x)*F^4 + (2*x^3-7*x^2+x-1)*F^3 - (x^3-10*x^2+3*x-3)*F^2 - (x^2+x+3)*F + 2*x + 1. - Nicholas R. Beaton, Jan 12 2026

a(n) ~ C * n^(-3/2) * d^n / sqrt(Pi), where d = 5.16207... is a root of 1 - 14*d + 7*d^2 - 6*d^3 + d^4 and C = 0.178902... is a root of 1 - 1520*C^2 + 477072*C^4 + 114892800*C^6 - 1225153536*C^8 - 89053982720*C^10 + 64192831488*C^12 - 11841044480*C^14 + 8540717056*C^16 = 0. - Nicholas R. Beaton, Jan 13 2026

In closed form, d = 1/(7/2 + (3*sqrt(5))/2 - sqrt(2*(11 + 5*sqrt(5)))), C = 1/(722*sqrt((2*(-1215 - 539*sqrt(5) + 180*sqrt(22 + 10*sqrt(5)) + 82*sqrt(110 + 50*sqrt(5))))/ (31057240 + 12603704*sqrt(5) - 16082460*sqrt(22 + 10*sqrt(5)) + 3207571*sqrt(110 + 50*sqrt(5)) - 495670*sqrt(2*(1275 + 737*sqrt(5) + 500*sqrt(22 + 10*sqrt(5)) - 164*sqrt(110 + 50*sqrt(5)))) - 82958*sqrt(10*(1275 + 737*sqrt(5) + 500*sqrt(22 + 10*sqrt(5)) - 164*sqrt(110 + 50*sqrt(5)))) + 2347305*sqrt(1100 - 433*sqrt(5) - 2400*sqrt(22 + 10*sqrt(5)) + 1076*sqrt(110 + 50*sqrt(5))) + 982609*sqrt(5*(1100 - 433*sqrt(5) - 2400*sqrt(22 + 10*sqrt(5)) + 1076*sqrt(110 + 50*sqrt(5))))))). - Vaclav Kotesovec, Jan 13 2026

Example

The length 3 inversion sequences are 000, 001, 002, 011, 012.
The length 4 inversion sequences are 0000, 0001, 0002, 0003, 0011, 0012, 0013, 0021, 0022, 0023, 0111, 0112, 0113, 0122, 0123.

Crossrefs

Cf. A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Sequence in context: A366096 A007317 A181768 * A385761 A279555 A153197

Adjacent sequences: A279551 A279552 A279553 * A279555 A279556 A279557

Keyword

nonn

Author

Megan A. Martinez, Dec 15 2016

Extensions

a(10)-a(11) from Alois P. Heinz, Feb 24 2017

a(12)-a(17) from Bert Dobbelaere, Dec 30 2018

a(18)-a(27) from Nicholas R. Beaton, Jan 12 2026

Status

approved