A328440 -id:A328440 - cp's OEIS frontend

A328357 Number of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.

1, 1, 2, 1, 4, 6, 36, 117, 804, 4266, 33768, 249144, 2289348, 21353472, 227212824, 2533824900, 30914509212, 398623158096, 5508014798052, 80377645583430, 1242697826967816, 20218588415853480, 346035438765576720, 6206862951272939550, 116518581654518098332

Offset: 0

Juan S. Auli, Oct 13 2019

nonn

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}. Alternatively, we can describe this as the set of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.

			The a(4)=4 length 4 inversion sequences avoiding the consecutive patterns 000, 001, 011, 012 are 0100, 0101, 0102, 0103.
The a(5)=6 length 5 inversion sequences are 01010, 01020, 01021, 01030, 01031, 01032.

Alois P. Heinz, Table of n, a(n) for n = 0..465
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

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=1..n))
    end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..24);  # Alois P. Heinz, Oct 14 2019

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, 1, n}]];
a[n_] :=  b[n, 0, False];
a /@ Range[0, 24] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n / n^(2*Pi/3^(3/2)), where c = 0.75844492121718325018323312623016463... - Vaclav Kotesovec, Oct 17 2019

Terms a(11)..a(16) from Joerg Arndt, Oct 14 2019

a(17)-a(24) from Alois P. Heinz, Oct 14 2019

A328358 Number of inversion sequences of length n avoiding the consecutive patterns 012, 021, 010, 120.

Original entry on oeis.org

1, 1, 2, 4, 10, 30, 100, 376, 1566, 7094, 34751, 182841, 1026167, 6112799, 38489481, 255204077, 1776046697, 12936265145, 98368170749, 779127467795, 6414876317675, 54802126603135, 484967246285755, 4438877330941077, 41963817964950737, 409224941931240185

Offset: 0

Juan S. Auli, Oct 13 2019

nonn

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 012, 021, 010, 120.

			The length 4 inversion sequences avoiding the consecutive patterns 012, 021, 010, 120 are 0000, 0110, 0001, 0011, 0111, 0002, 0112, 0022, 0003, 0113.

Alois P. Heinz, Table of n, a(n) for n = 0..578
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

b:= proc(n, x, t, c) option remember; `if`(n=0, 1, add(`if`(ix, max(0, c-1))), i=1..n))
    end:
a:= n-> b(n, 0, false, 2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 14 2019

b[n_, x_, t_, c_] := b[n, x, t, c] = If[n == 0, 1, Sum[If[i < x && t && c == 0, 0, b[n - 1, i, i != x, Max[0, c - 1]]], {i, 1, n}]];
a[n_] := b[n, 0, False, 2];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)

a(11)-a(25) from Alois P. Heinz, Oct 14 2019

A328441 Number of inversion sequences of length n avoiding the consecutive pattern 100.

Original entry on oeis.org

1, 1, 2, 6, 23, 109, 618, 4098, 31173, 267809, 2565520, 27120007, 313616532, 3938508241, 53381045786, 776672993274, 12074274033482, 199746500391688, 3503656507826887, 64951437702821877, 1268898555348831913, 26055882443142671038, 561050228044941209930, 12641053014560238560492, 297439800300471548183778

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

A length n inversion sequence e_1, e_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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 100.

The term a(n) also 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}. That is, a(n) also counts the inversion sequences of length n avoiding the consecutive pattern 110, see the Auli and Elizalde links.

			Note that a(4)=23. Indeed, of the 24 inversion sequences of length 4, the only one that does not avoid the consecutive pattern 100 is 0100.
Similarly, 0110 is the only inversion sequence of length 4 that does not avoid the consecutive pattern 110.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328442

# after Alois P. Heinz in A328357
b := proc(n, x, t) local i; option remember; `if`(n = 0, 1, add(`if`(t and x < i, 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);

i100[1] = 1; i100[2] = 2; i100[n_] := i100[n] = Sum[s100[n, k], {k, 0, n - 1}]; s100[n_, k_] := s100[n, k] = i100[n - 1] - Sum[s100[n - 2, j], {j, k + 1, n - 3}]; Flatten[{1, Table[i100[m], {m, 1, 25}]}] (* Vaclav Kotesovec, Oct 18 2019 *)

a(n) ~ n! * c / sqrt(n), where c = 2.428754692682297906864850201408427747198... - Vaclav Kotesovec, Oct 18 2019

A328433 Number of inversion sequences of length n avoiding the consecutive patterns 011 and 012.

Original entry on oeis.org

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

Juan S. Auli, Oct 16 2019

nonn

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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..464
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
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);

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 *)

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

A328437 Number of inversion sequences of length n avoiding the consecutive pattern 001.

Original entry on oeis.org

1, 1, 2, 4, 11, 42, 210, 1292, 9352, 77505, 722294, 7470003, 84854788, 1049924370, 14052654158, 202271440732, 3115338658280, 51118336314648, 890201500701303, 16397264064993185, 318505677099378561, 6506565509515408206, 139449260758011488550, 3128599281190613701180

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 001.

			The a(4)=11 length 4 inversion sequences avoiding the consecutive pattern 001 are 0000, 0100, 0110, 0120, 0101, 0111, 0121, 0102, 0122, 0103, and 0123.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
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);

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 *)

a(n) ~ n! * c / sqrt(n), where c = 0.549342310436989831962783548104445992522... - Vaclav Kotesovec, Oct 18 2019

A328439 Number of inversion sequences of length n avoiding the consecutive pattern 011.

Original entry on oeis.org

1, 1, 2, 5, 17, 75, 407, 2621, 19524, 165090, 1561900, 16345264, 187452475, 2337729329, 31497068553, 455930417721, 7056447326642, 116279714536838, 2032547040624336, 37563420959431569, 731810131489893185, 14989602024463575408, 322032777284323744894, 7240745954488939549295

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 011.

			The a(4)=17 length 4 inversion sequences avoiding the consecutive pattern 011 are 0000, 0100, 0010, 0020, 0120, 0001, 0101, 0021, 0121, 0002, 0102, 0012, 0003, 0103, 0013, 0023, and 0123.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
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);

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 *)

a(n) ~ n! * c / sqrt(n), where c = 1.3306953765239857433314976921138998977998... - Vaclav Kotesovec, Oct 19 2019

A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 684, 4554, 34192, 285558, 2624496, 26315990, 285828324, 3342566724, 41869664320, 559265742918, 7934746600620, 119162454310392, 1888417811354292, 31492626988890798, 551302582228438512, 10107905106374914860, 193700015975819881008, 3872391687779493752340, 80623321999146782133372

Offset: 0

Juan S. Auli, Oct 17 2019

nonn

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 210.

			Note that a(5)=118. Indeed, of the 120 inversion sequences of length 5, the only ones that do not avoid the consecutive patterns 210 are 00210 and 01210.

Vaclav Kotesovec, Table of n, a(n) for n = 0..460
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and x < i, 0, b(n - 1, i, x < i)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, n, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x < i, 0, b[n - 1, i, x < i]], {i, 0, n - 1}]];
a[n_] := b[n, n, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020, after Alois P. Heinz in A328357 *)

a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n * n^(2*Pi/3^(3/2)), where c = 0.24427562500895080639039917229089... - Vaclav Kotesovec, Oct 19 2019

A328429 Number of inversion sequences of length n avoiding the consecutive patterns 012, 101, 102, and 201.

Original entry on oeis.org

1, 1, 2, 5, 14, 46, 170, 691, 3073, 14809, 76666, 423886, 2490514, 15479614, 101389508, 697513653, 5025406212, 37819960947, 296618360520, 2419362514273, 20484053318220, 179723185666151, 1631519158000420, 15302546831928727, 148099068509673563

Offset: 0

Juan S. Auli, Oct 15 2019

nonn

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 012, 101, 102, and 201.

			The a(4)=14 length 4 inversion sequences avoiding the consecutive patterns 012, 101, 102, and 201 are 0000, 0100, 0010, 0110, 0020, 0001, 0011, 0111, 0021, 0002, 0112, 0022, 0003, and 0113.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and i <> x, 0, b(n-1, i, i b(n, -1, false):
seq(a(n), n = 0 .. 24);

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 *)

A328430 Number of inversion sequences of length n avoiding the consecutive patterns 001 and 012.

Original entry on oeis.org

1, 1, 2, 3, 7, 18, 70, 317, 1825, 11805, 88212, 727731, 6660103, 66377942, 718681969, 8376682083, 104703957902, 1395883946839, 19777652272297, 296686846198829, 4697959440255354, 78299282813403618, 1370127872827224359, 25114095425698971152, 481202765468970358153

Offset: 0

Juan S. Auli, Oct 15 2019

nonn

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 001 and 012.

			The a(4)=7 length 4 inversion sequences avoiding the consecutive patterns 001 and 012 are 0000, 0100, 0110, 0101, 0111, 0102, and 0103.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
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);

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 *)

A328431 Number of inversion sequences of length n avoiding the consecutive patterns 010, 021, 120, and 210.

Original entry on oeis.org

1, 1, 2, 5, 15, 53, 214, 960, 4701, 24873, 141147, 853641, 5472642, 37024569, 263342224, 1962835806, 15288074104, 124120865849, 1048092680689, 9186689045482, 83435365244510, 783923558286071, 7608398620990535, 76177574145052258, 785853360840424425

Offset: 0

Juan S. Auli, Oct 16 2019

nonn

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 010, 021, 120, and 210.

			The a(4)=15 length 4 inversion sequences avoiding the consecutive patterns 010, 021, 120, and 210 are 0000, 0110, 0001, 0011, 0111, 0002, 0012, 0112, 0022, 0122, 0003, 0013, 0113, 0023, and 0123.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A328357, A328358, A328429, A328430, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and i <> x, 0, b(n - 1, i, x < i)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, n, false):
seq(a(n), n = 0 .. 24);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i != x, 0, b[n - 1, i, x < i]], {i, 0, n - 1}]];
a[n_] := b[n, n, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020 after Alois P. Heinz in A328357 *)

cp's OEIS Frontend

A328357 Number of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.

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A328358 Number of inversion sequences of length n avoiding the consecutive patterns 012, 021, 010, 120.

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A328441 Number of inversion sequences of length n avoiding the consecutive pattern 100.

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A328433 Number of inversion sequences of length n avoiding the consecutive patterns 011 and 012.

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A328437 Number of inversion sequences of length n avoiding the consecutive pattern 001.

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A328439 Number of inversion sequences of length n avoiding the consecutive pattern 011.

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A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.

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