Thomas Zaslavsky - cp's OEIS frontend

A385523 Numbers that are not cycle counts of inseparable graphs.

2, 4, 5, 8, 9, 16

Offset: 1

Thomas Zaslavsky, Jul 01 2025

Since subdividing edges does not change the inseparability or the cycle count, the sequence holds equally for simple graphs and multigraphs.

Ryan McCulloch, Brendan D. McKay, Alireza Salahshoori, and Thomas Zaslavsky, The Cycle Counts of Graphs, arXiv:2507.02260 [math.CO], 2025.

Cf. A385524 (cubic graphs).

A385524 Numbers that are not cycle counts of inseparable cubic multigraphs.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 13, 16

Offset: 1

Thomas Zaslavsky, Jul 01 2025

Ryan McCulloch, Brendan D. McKay, Alireza Salahshoori, and Thomas Zaslavsky, The Cycle Counts of Graphs, arXiv:2507.02260 [math.CO], 2025.

Cf. A385523.

A379695 Number of equivalence classes of regular vines in dimension n.

Original entry on oeis.org

1, 1, 1, 2, 6, 40, 560, 17024

Offset: 1

Thomas Zaslavsky, Jan 03 2025

a(n) is also the number of non-isomorphic MAT-labelings of complete graph K_n. (Tran, Tran, Tsujie, 2024, Theorem 3.1).

D. Kurowicka and H. Joe, eds. Dependence Modeling. Vine Copula Handbook. World Scientific, Hackensack, NJ, 2011. (§10.3)

Hung Manh Tran, Tan Nhat Tran, and Shuhei Tsujie, Vines and MAT-labeled graphs, Séminaire Lotharingien de Combinatoire 91B (2024) Article #12, 12 pp.

A179061 Number of non-attacking placements of 6 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 720, 35280, 564480, 5080320, 31752000, 153679680, 614718720, 2120152320, 6492966480, 18036018000, 46172206080, 110279070720, 248127909120, 530024705280, 1081683072000, 2120098821120, 4008311833680

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Column k=5 of A144084.

Cf. A179060 (5 rooks), A179062 (7 rooks).

a(n) = 6! * binomial(n, 6)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 6! * binomial(n, 6)^2.

G.f.: -720*x^6*(x^6+36*x^5+225*x^4+400*x^3+225*x^2+36*x+1) / (x-1)^13. - Colin Barker, Jan 08 2013

A179062 Number of non-attacking placements of 7 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 5040, 322560, 6531840, 72576000, 548856000, 3161410560, 14841066240, 59364264960, 208702494000, 659602944000, 1906252508160, 5104345559040, 12796310741760, 30287126016000, 68146033536000

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Column k=7 of A144084.

Cf. A179061 (6 rooks), A179063 (8 rooks).

7! Binomial[Range[30],7]^2 (* or *) LinearRecurrence[{15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1},{0,0,0,0,0,0,5040,322560,6531840,72576000,548856000,3161410560,14841066240,59364264960,208702494000},30] (* Harvey P. Dale, May 25 2017 *)

a(n) = 7! * binomial(n, 7)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 7!*binomial(n,7)^2.

G.f.: -5040*x^7*(x+1)*(x^6+48*x^5+393*x^4+832*x^3+393*x^2+48*x+1) / (x-1)^15. - Colin Barker, Jan 08 2013

A179063 Number of non-attacking placements of 8 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 40320, 3265920, 81648000, 1097712000, 9879408000, 66784798080, 363606122880, 1669619952000, 6678479808000, 23828156352000, 77203226580480, 230333593351680, 639815537088000, 1669577821632000, 4122835028928000

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Column k=8 of A144084.

Cf. A179062 (7 rooks), A179064 (9 rooks).

a(n) = 8!*binomial(n, 8)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 8!*binomial(n,8)^2.

G.f.: -40320*x^8*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1) / (x -1)^17. - Colin Barker, Jan 08 2013

A179059 Number of non-attacking placements of 4 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 24, 600, 5400, 29400, 117600, 381024, 1058400, 2613600, 5880600, 12269400, 24048024, 44717400, 79497600, 135945600, 224726400, 360561024, 563376600, 859685400, 1284221400, 1881864600, 2709885024, 3840540000, 5364060000

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Seth Chaiken, Christopher R. H. Hanusa and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Column k=4 of A144084.

Cf. A179058 (3 rooks), A179060 (5 rooks).

LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,0,24,600,5400,29400,117600,381024},40] (* Harvey P. Dale, Feb 19 2013 *)
a[n_] := If[n<4, 0, Coefficient[n!*LaguerreL[n, x], x, n-4] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)

a(n) = 4! * binomial(n, 4)^2; \\ Andrew Howroyd, Feb 13 2018

a(n) = 4! * binomial(n, 4)^2.
From Colin Barker, Jan 08 2013: (Start)
a(n) = (n^2*(-6+11*n-6*n^2+n^3)^2)/24.
G.f.: -24*x^4*(x^4 +16*x^3 +36*x^2 +16*x +1) / (x -1)^9.
(End)
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=4} 1/a(n) = (20*Pi^2 - 197)/9.
Sum_{n>=4} (-1)^n/a(n) = (64*log(2) - 44)/9. (End)

A179064 Number of non-attacking placements of 9 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 362880, 36288000, 1097712000, 17563392000, 185513328000, 1454424491520, 9090153072000, 47491411968000, 214453407168000, 857813628672000, 3096707199505920, 10237048593408000, 31350961317312000

Offset: 1

Thomas Zaslavsky, Jun 28 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Column k=9 of A144084.

Cf. A179063 (8 rooks), A179065 (10 rooks).

a(n) = 9! * binomial(n, 9)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 9!*binomial(n,9)^2.

G.f.: -362880*x^9*(x +1)*(x^8 +80*x^7 +1216*x^6 +5840*x^5 +10036*x^4 +5840*x^3 +1216*x^2 +80*x +1) / (x -1)^19. - Colin Barker, Jan 08 2013

A179065 Number of non-attacking placements of 10 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 439084800, 15807052800, 296821324800, 3636061228800, 32724551059200, 232707918643200, 1372501805875200, 6948290392243200, 30967071995059200, 123868287980236800

Offset: 1

Thomas Zaslavsky, Jun 28 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Column k=10 of A144084.

Cf. A179064 (9 rooks).

a(n) = 10! * binomial(n, 10)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 10! * binomial(n, 10)^2.

A179060 Number of non-attacking placements of 5 rooks on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 120, 4320, 52920, 376320, 1905120, 7620480, 25613280, 75271680, 198764280, 480960480, 1082161080, 2289530880, 4594961280, 8809274880, 16225246080, 28844881920, 49689816120, 83217546720, 135870624120

Offset: 1

Thomas Zaslavsky, Jun 27 2010

Andrew Howroyd, Table of n, a(n) for n = 1..200
Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Column k=5 of A144084.

Cf. A179059 (4 rooks), A179061 (6 rooks).

a[n_] := If[n<5, 0, Coefficient[n!*LaguerreL[n, x], x, n-5] // Abs];
Array[a, 30] (* Jean-François Alcover, Jun 14 2018, after A144084 *)

a(n) = 5! * binomial(n, 5)^2 \\ Andrew Howroyd, Feb 13 2018

a(n) = 5! * binomial(n, 5)^2.

G.f.: -120*x^5*(x+1)*(x^4+24*x^3+76*x^2+24*x+1) / (x-1)^11. - Colin Barker, Jan 08 2013

cp's OEIS Frontend

User: Thomas Zaslavsky

A385523 Numbers that are not cycle counts of inseparable graphs.

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A385524 Numbers that are not cycle counts of inseparable cubic multigraphs.

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A379695 Number of equivalence classes of regular vines in dimension n.

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A179061 Number of non-attacking placements of 6 rooks on an n X n board.

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A179062 Number of non-attacking placements of 7 rooks on an n X n board.

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A179063 Number of non-attacking placements of 8 rooks on an n X n board.

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A179059 Number of non-attacking placements of 4 rooks on an n X n board.

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A179064 Number of non-attacking placements of 9 rooks on an n X n board.

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A179065 Number of non-attacking placements of 10 rooks on an n X n board.

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A179060 Number of non-attacking placements of 5 rooks on an n X n board.

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