A328357 -id:A328357 - cp's OEIS frontend

A328434 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, 201, and 210.

1, 1, 2, 6, 21, 81, 346, 1630, 8350, 45958, 269815, 1681285, 11071336, 76743040, 558062437, 4244853573, 33687390663, 278296576327, 2388351295760, 21254019548162, 195801111412320, 1864508416302520, 18326903140310011, 185711672802101781, 1937795878138303715

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 101, 102, 201, and 210.

			Note that a(4)=21. Indeed, of the 24 inversion sequences of length 4, the only ones that do not avoid the consecutive patterns 101, 102, 201, and 210 are 0101, 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, A328430, A328431, A328432, A328433, 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 and x>-1)), 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 && x > -1]], {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 *)

A328435 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, and 201.

Original entry on oeis.org

1, 1, 2, 6, 21, 83, 368, 1814, 9837, 58095, 370499, 2534374, 18493023, 143280489, 1173971656, 10136279104, 91936857611, 873547634921, 8673546319685, 89796095349193, 967384904147690, 10825116242427973, 125613702370667158, 1509222589338456874, 18748890945849736182

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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 101, 102, and 201.

			Note that a(4)=21. Indeed, of the 24 inversion sequences of length 4, the only ones that do not avoid the consecutive patterns 101, 102, and 201 are 0101, 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, A328430, A328431, A328432, A328433, A328434, 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 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);

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x < i, 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 *)

A328436 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 001.

Original entry on oeis.org

1, 1, 2, 3, 9, 37, 190, 1181, 8564, 70914, 659810, 6811371, 77232836, 953969548, 12747856402, 183218649413, 2818050980941, 46182485773217, 803323102085452, 14781372445602234, 286838921699435184, 5854404018902152208, 125367868007259046305, 2810511319383912299122

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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 001.

			The a(4)=9 length 4 inversion sequences avoiding the consecutive patterns 000 and 001 are 0100, 0110, 0120, 0101, 0121, 0102, 0122, 0103, 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, A328431, A328432, A328433, A328434, A328435, 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 *)

A328438 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 011.

Original entry on oeis.org

1, 1, 2, 4, 13, 57, 304, 1937, 14315, 120264, 1131896, 11794453, 134774963, 1675630582, 22516745452, 325188337067, 5022796990606, 82620491929333, 1441894214312037, 26609607869036180, 517741915593936360, 10592513721179374467, 227325651424365263577, 5106351205789851629476

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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 011.

			The a(4)=13 length 4 inversion sequences avoiding the consecutive patterns 000 and 011 are 0100, 0010, 0020, 0120, 0101, 0021, 0121, 0102, 0012, 0103, 0013, 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, A328431, A328432, A328433, A328434, A328435, A328436, A328437, 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 *)

A328440 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 100.

Original entry on oeis.org

1, 1, 2, 5, 18, 81, 448, 2920, 21955, 186981, 1779170, 18706222, 215364181, 2694650157, 36408144034, 528302958022, 8193953571315, 135277259197031, 2368556730208679, 43838335667451773, 855200666797199814, 17538187897491897945, 377199969925672569364, 8489656058119117230574

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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 110, see the Auli and Elizalde reference.

			The a(4)=18 length 4 inversion sequences avoiding the consecutive patterns 000 and 100 are 0010, 0110, 0020, 0120, 0101, 0011, 0021, 0121, 0102, 0012, 0112, 0022, 0122, 0103, 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, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328441, A328442.

# 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, 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 && x <= i, 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 *)

A328409 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i <= j >= k.

Original entry on oeis.org

1, 1, 2, 3, 6, 16, 57, 245, 1248, 7151, 46104, 325560, 2523437, 21106494, 190806861, 1842347541, 19018910502, 208088481921, 2414462433024, 29512737830802, 380156646308541, 5133381861786182, 72678441538790901, 1074324277172134786, 16581261996774703606

Offset: 0

Alois P. Heinz, Oct 14 2019

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 3: 000, 010, 011.
a(4) = 6: 0000, 0101, 0102, 0103, 0110, 0111.
a(5) = 16: 00000, 01010, 01011, 01020, 01021, 01022, 01030, 01031, 01032, 01033, 01101, 01102, 01103, 01104, 01110, 01111.

Alois P. Heinz, Table of n, a(n) for n = 0..485
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A000108, A328357, A328358, A328425, A328491.

b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`((i>j
     xor t) and c=0, 0, b(n-1, i, is(i b(n, 0, true, 2):
seq(a(n), n=0..24);

b[n_, j_, t_, c_] := b[n, j, t, c] = If[n == 0, 1, Sum[If[Xor[i>j, t] && c == 0, 0, b[n - 1, i, iJean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

a(n) ~ n! * c * 2^n * n^(Pi/4 - 1/2) / Pi^n, where c = 0.52096414784... - Vaclav Kotesovec, Oct 31 2019

A328425 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i < j > k or i >= j <= k.

Original entry on oeis.org

1, 1, 2, 4, 11, 36, 142, 647, 3383, 19816, 129162, 923279, 7201951, 60720996, 551268926, 5352973967, 55430433719, 609033864160, 7083303687843, 86864585123112, 1120997775904467, 15176639841694385, 215196709973260722, 3187766448289854016, 49262381105608795771

Offset: 0

Alois P. Heinz, Oct 15 2019

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 4: 000, 001, 002, 010.
a(5) = 11: 0000, 0001, 0002, 0003, 0010, 0020, 0021, 0100, 0101, 0102, 0103.
a(6) = 36: 00000, 00001, 00002, 00003, 00004, 00010, 00020, 00021, 00030, 00031, 00032, 00100, 00101, 00102, 00103, 00104, 00200, 00201, 00202, 00203, 00204, 00211, 00212, 00213, 00214, 01000, 01001, 01002, 01003, 01004, 01010, 01020, 01021, 01030, 01031, 01032.

Alois P. Heinz, Table of n, a(n) for n = 0..485
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A000108, A328357, A328358, A328409, A328491.

b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`((ij), max(0, c-1))), i=1..n))
    end:
a:= n-> b(n, 0, true, 2):
seq(a(n), n=0..24);

b[n_, j_, t_, c_] := b[n, j, t, c] = If[n == 0, 1, Sum[If[Xor[i < j, t] && c == 0, 0, b[n - 1, i, i > j, Max[0, c - 1]]], {i, 1, n}]];
a[n_] := b[n, 0, True, 2];
a /@ Range[0, 24] (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

a(n) ~ n! * c * 2^n * n^(Pi/4 - 1/2) / Pi^n, where c = 1.60233729528... - Vaclav Kotesovec, Oct 31 2019

A328491 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j <= k or i <= j > k.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 32, 89, 592, 2402, 19072, 101866, 939136, 6221228, 65291264, 516212409, 6075261184, 55812055946, 727912302592, 7618369901774, 109058247342080, 1280820543489044, 19965414947799040, 259988000952099210, 4383593333171363840, 62680335913868539796

Offset: 0

Alois P. Heinz, Oct 16 2019

nonn

			a(0) = 1: the empty sequence.
a(1) = 1: 0.
a(2) = 2: 00, 01.
a(3) = 1: 010.
a(4) = 4: 0100, 0101, 0102, 0103.
a(5) = 6: 01010, 01020, 01021, 01030, 01031, 01032.
a(6) = 32: 010100, 010101, 010102, 010103, 010104, 010105, 010200, 010201, 010202, 010203, 010204, 010205, 010211, 010212, 010213, 010214, 010215, 010300, 010301, 010302, 010303, 010304, 010305, 010311, 010312, 010313, 010314, 010315, 010322, 010323, 010324, 010325.

Alois P. Heinz, Table of n, a(n) for n = 0..485
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A000108, A000142, A000225, A001250, A328357, A328358, A328409, A328425.

b:= proc(n, j, t, c) option remember; `if`(n=0, 1, add(`if`(c=0 and
      (i>j xor t), 0, b(n-1, i, is(i<=j), max(0, c-1))), i=1..n))
    end:
a:= n-> b(n, 0, true, 2):
seq(a(n), n=0..27);

b[n_, j_, t_, c_] := b[n, j, t, c] = If[n == 0, 1, Sum[If[Xor[i > j, t] && c == 0, 0, b[n - 1, i, i <= j, Max[0, c - 1]]], {i, 1, n}]];
a[n_] := b[n, 0, True, 2];
a /@ Range[0, 27] (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

a(n) is odd <=> n in { A000225 }.

a(n) ~ n! * c * 2^n / Pi^n, where c = 0.292816379603485707589209784583144390652038770449692132953726209770208058... - Vaclav Kotesovec, Oct 17 2019

A326308 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i < j > k.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 26, 85, 476, 2171, 14905, 87153, 708825, 5053464, 47514180, 399542814, 4264132468, 41306091312, 493337571005, 5408829555639, 71476985762027, 874870165668858, 12673922434134249, 171294209823727623, 2699365743596908540, 39925463781029750810

Offset: 0

Alois P. Heinz, Oct 17 2019

nonn

			a(6) = 26: 010101, 010102, 010103, 010104, 010105, 010201, 010202, 010203, 010204, 010205, 010212, 010213, 010214, 010215, 010301, 010302, 010303, 010304, 010305, 010312, 010313, 010314, 010315, 010323, 010324, 010325.

Alois P. Heinz, Table of n, a(n) for n = 0..486
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A000108, A000142, A000225, A001250, A328357, A328358, A328409, A328425, A328491, A326412.

b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add(
      `if`(c>0 or i>j and t or ij), max(0, c-1)), 0), i=1..n))
    end:
a:= n-> b(n, 0, true$2, 2):
seq(a(n), n=0..25);

b[n_, j_, t_, u_, c_] := b[n, j, t, u, c] = If[n == 0, 1, Sum[If[c>0 || i>j && t || ij, Max[0, c-1]], 0], {i, 1, n}]];
a[n_] := b[n, 0, True, True, 2];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)

a(n) ~ n! * c * 2^n / (Pi^n * sqrt(n)), where c = 1.0215796642504649172542599982453320786973706265645819484... - Vaclav Kotesovec, Oct 31 2019

A326412 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i >= j <= k or i <= j >= k.

Original entry on oeis.org

1, 1, 2, 5, 17, 69, 330, 1797, 11028, 74932, 559351, 4540088, 39840318, 375421225, 3782383945, 40548234374, 460956742449, 5536790753853, 70077462043662, 931945968071778, 12993337101354500, 189485727877247991, 2884989393948284323, 45772604755492432599

Offset: 0

Alois P. Heinz, Oct 17 2019

nonn

			a(4) = 17: 0000, 0001, 0002, 0003, 0010, 0011, 0020, 0021, 0022, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113.

Alois P. Heinz, Table of n, a(n) for n = 0..483
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Cf. A000108, A000142, A000225, A001250, A328357, A328358, A328409, A328425, A328491, A326308.

b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add(
      `if`(c>0 or i>=j and t or i<=j and u, b(n-1, i,
        is(i<=j), is(i>=j), max(0, c-1)), 0), i=1..n))
    end:
a:= n-> b(n, 0, true$2, 2):
seq(a(n), n=0..25);

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

a(n) ~ n! * c * 2^n * n^((Pi+1)/2) / Pi^n, where c = 0.0662002484840446134... - Vaclav Kotesovec, Oct 31 2019

cp's OEIS Frontend

A328434 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, 201, and 210.

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A328435 Number of inversion sequences of length n avoiding the consecutive patterns 101, 102, and 201.

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A328436 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 001.

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A328438 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 011.

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A328440 Number of inversion sequences of length n avoiding the consecutive patterns 000 and 100.

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A328409 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i <= j >= k.

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A328425 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i < j > k or i >= j <= k.

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A328491 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j <= k or i <= j > k.

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