A263777 -id:A263777 - cp's OEIS frontend

A279552 Number of length n inversion sequences avoiding the patterns 000 and 010.

1, 1, 2, 4, 10, 29, 95, 345, 1376, 5966, 27886, 139608, 744552, 4210191, 25140790, 157981820, 1041480482, 7183374125, 51711299169, 387683162541, 3020997261596, 24424884853963, 204559337781097, 1772011400733378, 15855597322378302, 146360032952969570

Offset: 0

Megan A. Martinez, Dec 15 2016

nonn

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= e_k and e_i = e_k. This is the same as the set of length n inversion sequences avoiding 000 and 010.

			For n=3, the inversion sequences are 001, 002, 011, 012.
For n=4, the inversion sequences are 0011, 0012, 0013, 0021, 0022, 0023, 0112, 0113, 0122, 0123.

Benjamin Testart, Table of n, a(n) for n = 0..200
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Benjamin Testart, Inversion sequences avoiding the pattern 010, arXiv:2212.07222 [math.CO], 2022.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.

Cf. A263777, A263778, A263779, A263780, A279551, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

a(10)-a(20) from Alois P. Heinz, Feb 23 2017

a(21) onwards from Benjamin Testart, Feb 01 2023

A279561 Number of length n inversion sequences avoiding the patterns 101, 102, 201, and 210.

Original entry on oeis.org

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

Offset: 0

Megan A. Martinez, Jan 17 2017

nonn

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 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 e_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 021 and 120.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..1664
Shane Chern, On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma, arXiv:2006.04318 [math.CO], 2020.
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

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.

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:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 18 2017

a[n_] := 1 + Sum[Binomial[2i, i-1], {i, 0, n-1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 28 2017 *)

a(n) = 1 + Sum_{i=1..n-1} binomial(2i, i-1).
a(n) = 1 + A057552(n-2).
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

A279551 Number of length n inversion sequences avoiding the patterns 000, 010, 110, and 120.

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 79, 247, 816, 2822, 10158, 37875, 145695, 576288, 2337412, 9698820, 41089107, 177424188, 779699793, 3482575169, 15791709187, 72621800171, 338388714955, 1596314968112, 7618218238583, 36756086159343, 179176803145900, 882002961543492

Offset: 0

Megan A. Martinez, Dec 15 2016

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= e_k. This is the same as the set of length n inversion sequences avoiding 000, 010, 110, and 120.

			For n=3, the inversion sequences are 001, 002, 011, 012.
For n=4, the inversion sequences are 0011, 0012, 0013, 0021, 0022, 0023, 0112, 0113, 0122, 0123.

Nicholas R. Beaton, Table of n, a(n) for n = 0..65
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Cf. A263777, A263778, A263779, A263780, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Typo in comment corrected and a(10)-a(27) added by Alois P. Heinz, Feb 22 2017

A279553 Number of length n inversion sequences avoiding the patterns 110, 210, 120, 201, and 010.

Original entry on oeis.org

1, 1, 2, 5, 15, 50, 178, 663, 2552, 10071, 40528, 165682, 686151, 2872576, 12137278, 51690255, 221657999, 956265050, 4147533262, 18074429421, 79102157060, 347519074010, 1532070899412, 6775687911920, 30052744139440, 133649573395725, 595816470717728

Offset: 0

Megan A. Martinez, Dec 15 2016

nonn

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 e_k and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 010, 110, 120, 201, and 210.

It can be shown that this sequence also counts the length n inversion sequences with no entries e_i, e_j, e_k (where i e_j and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 010, 100, 120, 201, and 210.

			The length 3 inversion sequences avoiding (110, 210, 120, 201, 010) are 000, 001, 002, 011, 012.
The length 4 inversion sequences avoiding (110, 210, 120, 201, 010) are 0000, 0001, 0002, 0003, 0011, 0012, 0013, 0021, 0022, 0023, 0111, 0112, 0113, 0122, 0123.

Alois P. Heinz, Table of n, a(n) for n = 0..1489
Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Cf. A263777, A263778, A263779, A263780, A279551, A279552, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 5][n+1],
      ((12*(n-1))*(182*n^3-1659*n^2+4628*n-3756)*a(n-1)
      -(4*(91*n^4-1057*n^3+3812*n^2-4046*n-906))*a(n-2)
      +(6*(n-4))*(182*n^3-1659*n^2+4901*n-4630)*a(n-3)
      -(4*(n-4))*(n-5)*(91*n^2-511*n+690)*a(n-4))
       /(5*n*(n-1)*(91*n^2-693*n+1292)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 22 2017

a[n_] := a[n] = If[n < 4, {1, 1, 2, 5}[[n + 1]], ((12*(n - 1))*(182*n^3 - 1659*n^2 + 4628*n - 3756)*a[n - 1] - (4*(91*n^4 - 1057*n^3 + 3812*n^2 - 4046*n - 906))*a[n - 2] + (6*(n - 4))*(182*n^3 - 1659*n^2 + 4901*n - 4630)*a[n - 3] - (4*(n - 4))*(n - 5)*(91*n^2 - 511*n + 690)*a[n - 4]) / (5*n*(n - 1)*(91*n^2 - 693*n + 1292))]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

seq(N) = my(x='x+O('x^N)); Vec(1+serreverse((-x^3+x^2+x)/(2*x^2+3*x+1)));
seq(27) \\ Gheorghe Coserea, Jul 11 2018

G.f.: 1 + Series_Reversion(x*A094373(-x)). - Gheorghe Coserea, Jul 11 2018
a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 4.730576939379622099763633264641101585205420756515858657461873... is the greatest real root of the equation 4 - 12*d + 4*d^2 - 24*d^3 + 5*d^4 = 0 and c = 0.3916760466183576202289779130261876915536170330427700961416097... is the positive real root of the equation -5 - 64*c^2 - 33728*c^4 + 209664*c^6 + 93184*c^8 = 0. - Vaclav Kotesovec, Jul 12 2018
D-finite with recurrence: 45*n*(n-1)*a(n) -4*(n-1)*(49*n-66)*a(n-1) +2*(-25*n^2+157*n-264)*a(n-2) +2*(-70*n^2+445*n-714)*a(n-3) -4*(n-6)*(n-13)*a(n-4) -4*(n-6)*(2*n-17)*a(n-5) +8*(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Feb 21 2020

a(10)-a(16) from Lars Blomberg, Feb 02 2017

a(17)-a(26) from Alois P. Heinz, Feb 22 2017

A279556 Number of length n inversion sequences avoiding the patterns 010, 110, and 120.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 190, 759, 3206, 14180, 65203, 309998, 1517330, 7619541, 39145113, 205261890, 1096393056, 5955598301, 32852080738, 183797522935, 1041802426740, 5977047039743, 34679912608313, 203345277644481, 1204104271508239, 7196256426157901

Offset: 0

Megan A. Martinez, Dec 16 2016

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 e_k and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 010, 110, and 120.

			The length 3 inversion sequences avoiding (010, 110, 120) are 000, 001, 002, 011, 012
The length 4 inversion sequences avoiding (010, 110, 120) are 0000, 0001, 0002, 0003, 0011, 0012, 0013, 0021, 0022, 0023, 0111, 0112, 0113, 0122, 0123.

Nicholas R. Beaton, Table of n, a(n) for n = 0..54
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.

Cf. A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

a(10)-a(11) from Alois P. Heinz, Feb 24 2017
a(12)-a(16) from Bert Dobbelaere, Dec 30 2018
a(17)-a(25) from Nicholas R. Beaton, Aug 29 2025

A279560 Number of length n inversion sequences avoiding the patterns 100, 210, 201, and 102.

Original entry on oeis.org

1, 1, 2, 6, 21, 76, 277, 1016, 3756, 13998, 52554, 198568, 754316, 2878552, 11027384, 42384412, 163372325, 631290168, 2444700421, 9485463044, 36866810877, 143508889270, 559399074443, 2183269032876, 8530724152279, 33366805383326, 130633854520329, 511889287682280

Offset: 0

Megan A. Martinez, Jan 17 2017

nonn

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 e_j and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 100, 210, 201, and 102.

			The length 4 inversion sequences avoiding (100, 210, 201, 102) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.

Alois P. Heinz, Table of n, a(n) for n = 0..1665
Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Cf. A000108, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

a:= proc(n) option remember; `if`(n<4, n!,
     ((6*(9*n^4-61*n^3+100*n^2+52*n-140))*a(n-1)
     -(3*(3*n-8))*(9*n^3-38*n^2+3*n+70)*a(n-2)
     +(2*(2*n-7))*(9*n^3-31*n^2-2*n+60)*a(n-3))
      / ((9*n^3-58*n^2+87*n+22)*n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 24 2017

a[0] = 1; a[n_] := Binomial[2n-2, n-1] + Sum[(4i Binomial[2i+1, i+1]) / ((i+2)(i+3)), {k, 2, n-2}, {i, 1, k-1}]; Array[a, 30, 0] (* Jean-François Alcover, Nov 06 2017 *)

a(n) = if (n==0, 1, binomial(2*n-2,n-1) + sum(k=2, n-2, sum(i=1,k-1, sum(u=1, i, sum(d=0, u-1, ((i-d+1)/(i+1)*binomial(i+d,d))))))); \\ Michel Marcus, Jan 18 2017

a(n) = binomial(2n-2,n-1) + Sum_{k=2..n-2} Sum_{i=1..k-1} Sum_{u=1..i} Sum_{d=0..u-1} ((i-d+1)/(i+1)*binomial(i+d,d)) for n>0, a(0)=1.

a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 07 2021

More terms from Michel Marcus, Jan 18 2017

A279562 Number of length n inversion sequences avoiding the patterns 100, 102, and 201.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 299, 1176, 4729, 19378, 80667, 340260, 1451277, 6248758, 27124703, 118576648, 521574769, 2306766426, 10251761219, 45759404076, 205050758165, 922104978430, 4160045001703, 18823187479504, 85400356099001, 388422301113250, 1770695668597643, 8089198184655732, 37027394471695197

Offset: 0

Megan A. Martinez, Feb 09 2017

nonn

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 e_j <= e_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 100, 102, and 201.

			The length 4 inversion sequences avoiding (100, 102, 201) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.

Nathan J. Britt, Table of n, a(n) for n = 0..1000
Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016-2018.

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

G.f.: (2 + x - 10*x^2 + 4*x^3 - (2-3*x)*(1 - 4*x - 4*x^2)^(1/2)) / (8*x*(1 - x)^2). - Nathan J. Britt, Jun 08 2025

a(n) ~ c * (2 + 2*sqrt(2))^n / n^(3/2), where c = 0.40413545332026258682681691461076303199449216224437... - Nathan J. Britt, Jun 08 2025

a(10)-a(12) from Alois P. Heinz, Feb 24 2017
a(13)-a(17) from Bert Dobbelaere, Dec 30 2018
More terms from Nathan J. Britt, Jun 08 2025

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

Original entry on oeis.org

1, 1, 2, 6, 22, 85, 328, 1253, 4754, 17994, 68158, 258808, 985906, 3768466, 14451386, 55585014, 214377618, 828795169, 3211030684, 12464308997, 48465092366, 188733879657, 735977084412, 2873525548315, 11231884145434, 43947466923095, 172115939825516

Offset: 0

Megan A. Martinez, Feb 09 2017

nonn

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 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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..1664
Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

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.

a:= proc(n) option remember; `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)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 22 2017

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 *)

a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2021

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

a(10)-a(26) from Alois P. Heinz, Feb 22 2017

A279567 Number of length n inversion sequences avoiding the patterns 100, 110, 120, and 210.

Original entry on oeis.org

1, 1, 2, 6, 21, 82, 343, 1509, 6893, 32419, 156058, 765578, 3815062, 19263736, 98368919, 507197436, 2637242188, 13814247530, 72834238423, 386244387688, 2058933104170, 11026807340592, 59304897232442, 320181600386661, 1734685419170666, 9428340999504441

Offset: 0

Megan A. Martinez, Feb 09 2017

nonn

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= e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 110, 120, and 210.

			The length 4 inversion sequences avoiding (100, 110, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279568, A279569, A279570, A279571, A279572, A279573.

b:= proc(n, i, m) option remember; `if`(n=0, 1, add((h->
      b(n-1, i-h+1, max(m, j)-h))(max(0, min(m-1, j))), j=1..i))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 23 2017

b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[b[n-1, i-#+1, Max[m, j]-#]& @ Max[0, Min[m-1, j]], {j, 1, i}]]; a[n_] := b[n, 1, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)

a(n) ~ c * (1 + sqrt(2))^(2*n) / n^(3/2), where c = 0.066085708825649431003670013119332303648755519420440375... - Vaclav Kotesovec, Oct 07 2021

a(10)-a(25) from Alois P. Heinz, Feb 23 2017

A279568 Number of length n inversion sequences avoiding the patterns 110, 120, 201, and 210.

Original entry on oeis.org

1, 1, 2, 6, 22, 90, 396, 1833, 8801, 43441, 219092, 1124201, 5850414, 30805498, 163824559, 878655117, 4747341879, 25815026491, 141173582016, 775920816789, 4283833709457, 23746640019657, 132116647765569, 737485227605338, 4129174120158569, 23183379592361839

Offset: 0

Megan A. Martinez, Feb 21 2017

nonn

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 e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 110, 120, 201, and 210.

It was shown that a_n also counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_j and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 120, 201, and 210.

			The length 4 inversion sequences avoiding (110, 120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
The length 4 inversion sequences avoiding (100, 120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0121, 0122, 0123.

Alois P. Heinz, Table of n, a(n) for n = 0..1294
Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279569, A279570, A279571, A279572, A279573.

b:= proc(n, i, l) option remember; `if`(n=0, 1, add((h->
      b(n-1, i-h+2, j-h+1))(max(1, `if`(j=l, 0, l))), j=1..i))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 23 2017

b[n_, i_, l_] := b[n, i, l] = If[n == 0, 1, Sum[b[n-1, i-#+2, j-#+1]& @ Max[1, If[j == l, 0, l]], {j, 1, i}]]; a[n_] :=  b[n, 1, 1];  Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 5.98041772076926677236919875200507... is the positive root of the equation -32 - 195*d - 12*d^2 - 112*d^3 + 20*d^4 = 0 and c = 0.1056946795054351807407212356928404107733262398133039312067247126343... - Vaclav Kotesovec, Oct 07 2021

a(10)-a(25) from Alois P. Heinz, Feb 23 2017

cp's OEIS Frontend

A279552 Number of length n inversion sequences avoiding the patterns 000 and 010.

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A279561 Number of length n inversion sequences avoiding the patterns 101, 102, 201, and 210.

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A279551 Number of length n inversion sequences avoiding the patterns 000, 010, 110, and 120.

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A279553 Number of length n inversion sequences avoiding the patterns 110, 210, 120, 201, and 010.

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A279556 Number of length n inversion sequences avoiding the patterns 010, 110, and 120.

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A279560 Number of length n inversion sequences avoiding the patterns 100, 210, 201, and 102.

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A279562 Number of length n inversion sequences avoiding the patterns 100, 102, and 201.

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

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