Marni Mishna - cp's OEIS frontend

A374842 Number of 7-regular labeled graphs on 2n nodes.

1, 0, 0, 0, 1, 286884, 480413921130, 1803595358964773088, 15138592322753242235338875, 271849772205948458085090804526392, 9883018890803233316233360724489799227748, 689121157937951859333538097288863665976145304960

Offset: 0

Marni Mishna, Jul 23 2024

nonn

These entries are generated by a linear recurrence.

			For example, for n=4, a(4)=1 indicates that there is a single 7-regular graph on 2n=8 vertices. Specifically, this is the complete graph.

Marni Mishna, Table of n, a(n) for n = 0..89
Frédéric Chyzak and Marni Mishna, Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction-based approach, arXiv:2406.04753 [math.CO], 2024.

Alternating terms of column k=7 of A059441.

Cf. A165628 (unlabeled case).

A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^2 = with respect to the generating set {(x,1), (1,y)}.

Original entry on oeis.org

1, 2, 12, 92, 792, 7302, 70464, 702536, 7178568, 74771570, 790906012, 8472417384, 91724327928, 1001987961834, 11030476949952, 122247789508992, 1362840516623944, 15272530735735338, 171946029518128956, 1943927810200670820, 22059590401383177792, 251183781609841838444

Offset: 0

Marni Mishna, May 22 2018

nonn

The number of paths on the Cayley graph of ((Z/5Z)^*2, {(x,1), (1,y)} which start and end at the identity.

a(n) is the number of words of length 5n over the alphabet {x,y} that reduce to the empty string upon iteratively removing factors of x^5 and y^5. For n=2, the a(2)=12 words of length 10 that reduce to the empty string are x^10, y^10, x^i y^5 x^(5-i), for i=1..5, and y^i x^5 y^(5-i), for i=1..5.

J. P. Bell and M. J. Mishna, On the Complexity of the Cogrowth Sequence, arXiv:1805.08118 [math.CO], 2018-2019.

Related cogrowth sequences (Z/2Z)^*2: A126869; (Z/3Z)^*2: A047098; (Z/4Z)^*2: A107026.

terms = 22;
A[] = 0; Do[A[x] = (1 + 4 A[x] + 6 A[x]^2 + 4 A[x]^3 + A[x]^4 + 32 x A[x]^5)/(1 + A[x])^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 16 2018 *)

G.f.: A(x) satisfies 32*x*A(x)^5 - (A(x)-1)*(A(x)+1)^4 = 0.
a(n) satisfies the recurrence (2120000*(5*n+1))*(5*n+2)*(5*n+3)*(5*n+4)*a(n) + (250*(160980199*n^4 + 1129209134*n^3 + 2872721885*n^2 + 3155706646*n + 1267579560))*a(n+1) - (50*(109722203*n^4 + 959367613*n^3 + 3144281425*n^2 + 4572924587*n + 2485585548))*a(n+2) + (60*(4290021*n^4 + 51502996*n^3 + 243316306*n^2 + 532456081*n + 451079946))*a(n+3) - (3*(2673299*n^4 + 44756419*n^3 + 283571239*n^2 + 805783469*n + 866093430))*a(n+4) + (4008*(n+5))*(4*n+17)*(2*n+9)*(4*n+19)*a(n+5) = 0.
a(n) ~ 5^(5*n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(8*n - 3/2)). - Vaclav Kotesovec, Oct 24 2023
a(n) = binomial(5*n,n) - 3 * Sum_{k=0..n-1} binomial(5*n,k). - Seiichi Manyama, Apr 05 2024

A192867 Number of set partitions of {1, ..., n} that avoid enhanced 7-crossings (or enhanced 7-nestings).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644436, 190899266, 1382956734, 10480097431, 82863928963, 682058946982, 5832425824171, 51718812364549

Offset: 0

Marni Mishna, Jul 11 2011

nonn

			There are 27644437 partitions of 13 elements, but a(13)=27644436 because the partition {1,13}{2,12}{3,11}{4,10}{5,9}{6,8} {7} has an enhanced 7-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615, 2011
W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, math.CO/0501230

Cf. A000110, A108307, A192855, A192865, A192866.

A192866 Number of set partitions of {1, ..., n} that avoid enhanced 6-crossings (or enhanced 6-nestings).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213555, 27643388, 190878823, 1382610179, 10474709625, 82784673008, 680933897225, 5816811952612, 51505026270176

Offset: 0

Marni Mishna, Jul 11 2011

nonn

			There are 678570 partitions of 11 elements, but a(11)=678569 because the partition {1,11}{2,10}{3,9}{4,8}{5,9}{6} has an enhanced 6-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615, 2011
W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, math.CO/0501230

Cf. A000110, A108307, A192855, A192865.

A192865 Number of set partitions of {1,...,n} that avoid enhanced 5-crossings (or 5-nestings).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115945, 678012, 4205209, 27531954, 189486817, 1365888674, 10278272450, 80503198320, 654544093035, 5511256984436, 47950929125540

Offset: 0

Marni Mishna, Jul 11 2011

nonn

			There are 21147 partitions of 9 elements, but a(9)=21146 because the partition {1,9}{2,8}{3,7}{4, 6}{5} has an enhanced 5-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615, 2011
W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, math.CO/0501230
Juan B. Gil, Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705

Cf. A000110, A108307, A192855

A192855 Number of set partitions of {1, ..., n} that avoid enhanced 4-crossings (or enhanced 4-nestings).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4120, 20883, 113034, 648410, 3917021, 24785452, 163525976, 1120523114, 7947399981, 58172358642, 438300848329, 3391585460591, 26898763482122

Offset: 0

Marni Mishna, Jul 11 2011

			There are 877 partitions of 7 elements, but a(7)=51 because the partition {1,7}{2,6}{3,5}{4} has an enhanced 4-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011-2014.
W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018-2023. Also Discrete Mathematics (2019) Article 111705, doi:10.1016/j.disc.2019.111705.

Cf. A000110, A108307.

A192126 Number of set partitions of {1, ..., n} that avoid 5-nestings.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678530, 4212654, 27627153, 190624976, 1378972826, 10425400681, 82139435907, 672674215928, 5712423473216, 50193986895328, 455436027242590, 4259359394306331

Offset: 0

Marni Mishna, Jun 23 2011

a(n) is also equal to the number of set partitions of {1, ..., n} that avoid 5-crossings.

			There are 115975 partitions of 10 elements, but a(10)=115974 because the partition {1,10}{2,9}{3,8}{4,7}{5,6} has a 5-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011.
W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
M. Mishna and L. Yen, Set partitions with no k-nesting, arXiv:1106.5036 [math.CO], 2011-2012.

Cf. A108304, A108305.

A192128 Number of set partitions of {1, ..., n} that avoid 7-nestings.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899321, 1382958475, 10480139391, 82864788832, 682074818390, 5832698911490

Offset: 0

Marni Mishna, Jun 23 2011

This is equal to the number of set partitions of {1, ..., n} that avoid 7-crossings.

The first 14 terms coincide with terms of A000110. Without avoidance of 7-crossings, the two sequences would be identical. [Alexander R. Povolotsky, Sep 19 2011]

			There are 190899322 partitions of 14 elements, but a(14)=190899321 because the partition {1,14}{2,13}{3,12}{4,11}{5,10}{6,9}{7,8} has a 7-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011.
W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
M. Mishna and L. Yen, Set partitions with no k-nesting, arXiv:1106.5036 [math.CO], 2011-2012.

Cf. A000110.

A192127 Number of set partitions of {1, ..., n} that avoid 6-nestings.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213596, 27644383, 190897649, 1382919174, 10479355676, 82850735298, 681840170501, 5828967784989, 51665915664913, 473990899143781, 4493642492511044, 43959218211619150

Offset: 0

Marni Mishna, Jun 23 2011

nonn

This is equal to the number of set partitions of {1, ..., n} that avoid 6-crossings.

			There are 4213597 partitions of 12 elements, but a(12)=4213597 because the partition {1,12}{2,11}{3,10}{4,9}{5,8}{6,7} has a 6-nesting.

M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011.
W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
M. Mishna and L. Yen, Set partitions with no k-nesting, arXiv:1106.5036 [math.CO], 2011-2012.

Cf. A108304, A108305, A192126.

A108246 Number of labeled 2-regular graphs with no multiple edges, but loops are allowed (i.e., each vertex is endpoint of two (usual) edges or one loop).

Original entry on oeis.org

1, 1, 1, 2, 8, 38, 208, 1348, 10126, 86174, 819134, 8604404, 98981944, 1237575268, 16710431992, 242337783032, 3756693451772, 61991635990652, 1084943597643964, 20072853005524696, 391443701509660096, 8024999955144721256, 172544980412641191776

Offset: 0

Marni Mishna, Jun 17 2005

nonn

			a(3) = 2: {(1,2) (2,3) (1,3)}, {(1,1) (2,2) (3,3)}.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000985, A002137.
Binomial transform of A001205.
Row sums of A144161. - Alois P. Heinz, Jun 01 2009

b:= proc(n) option remember; if n=0 then 1 elif n<3 then 0 else (n-1) *(b(n-1) +b(n-3) *(n-2)/2) fi end: a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 12 2008

CoefficientList[Series[E^(-x^2/4+x/2)/Sqrt[1-x], {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 17 2012 *)

Linear recurrence satisfied by a(n): {a(2) = 1, a(0) = 1, (-n^2 - 3*n - 2)*a(n) + (4 + 2*n)*a(n+1) + (-2*n-6)*a(n+2) + 2*a(n+3), a(1) = 1}.
E.g.f.: exp(-t^2/4 + t/2)/sqrt(1-t). - Vladeta Jovovic, Aug 14 2006
a(n) ~ sqrt(2)*n^n/exp(n-1/4). - Vaclav Kotesovec, Oct 17 2012

More terms from Alois P. Heinz, Sep 12 2008

cp's OEIS Frontend

User: Marni Mishna

A374842 Number of 7-regular labeled graphs on 2n nodes.

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A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^*2 = * with respect to the generating set {(x,1), (1,y)}.

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A192867 Number of set partitions of {1, ..., n} that avoid enhanced 7-crossings (or enhanced 7-nestings).

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A192866 Number of set partitions of {1, ..., n} that avoid enhanced 6-crossings (or enhanced 6-nestings).

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A192865 Number of set partitions of {1,...,n} that avoid enhanced 5-crossings (or 5-nestings).

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A192855 Number of set partitions of {1, ..., n} that avoid enhanced 4-crossings (or enhanced 4-nestings).

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A192126 Number of set partitions of {1, ..., n} that avoid 5-nestings.

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A192128 Number of set partitions of {1, ..., n} that avoid 7-nestings.

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A192127 Number of set partitions of {1, ..., n} that avoid 6-nestings.

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A108246 Number of labeled 2-regular graphs with no multiple edges, but loops are allowed (i.e., each vertex is endpoint of two (usual) edges or one loop).

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A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^2 = with respect to the generating set {(x,1), (1,y)}.