A348821 -id:A348821 - cp's OEIS frontend

A348815 a(n) = number of chord labeled loopless diagrams by number of K_4.

0, 1, 134, 75843, 83002866, 158861646466, 490294453324924, 2292204611710892971, 15459367618357013402267, 144663877588996810362218074, 1819753109993633276315632934129, 29976383544377113242613349012354566, 632574848906117234957565158900144038734

Offset: 1

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..150
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 3.

Row 4 of A369923.

Cf. A190830, A348816, A348817; A003436, A348813, A348818, A348821.

a(11) onwards from Andrew Howroyd, Feb 05 2024

A369923 Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 31, 22, 1, 0, 1, 293, 1415, 134, 1, 0, 1, 3326, 140343, 75843, 866, 1, 0, 1, 44189, 20167651, 83002866, 4446741, 5812, 1, 0, 1, 673471, 3980871156, 158861646466, 55279816356, 276154969, 40048, 1, 0

Offset: 1

Andrew Howroyd, Feb 05 2024

Also, T(n,k) is the number of generalized chord labeled loopless diagrams with k parts of K_n. See the Krasko reference for a full definition.

			Array begins:
n\k| 1 2    3         4              5                    6 ...
---+-----------------------------------------------------------
 1 | 0 1    1         1              1                    1 ...
 2 | 0 1    4        31            293                 3326 ...
 3 | 0 1   22      1415         140343             20167651 ...
 4 | 0 1  134     75843       83002866         158861646466 ...
 5 | 0 1  866   4446741    55279816356     1450728060971387 ...
 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ...
 ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 51 antidiagonals)
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Rows 2..6 are A003436, A348813, A348815, A348818, A348821.
Column 3 is A197657, column 4 appears to be A209183(n)/2.
Cf. A322013 (without linearly adjacent elements equal), A322093.

T[n_, k_] := If[k == 1, 0, Expand[(-1)^(k (n + 1))/(k - 1)! n Hypergeometric1F1[1 - n, 2, x]^k x^(k - 1)] /. x^p_ :> p!] (* Eric W. Weisstein, Feb 20 2025 *)

\\ compare with A322013.
q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0)

A348813 a(n) = number of chord labeled loopless diagrams by number of K_3.

Original entry on oeis.org

0, 1, 22, 1415, 140343, 20167651, 3980871156, 1035707510307, 343866839138005, 141979144588872613, 71386289535825383386, 42954342000612934599071, 30482693813120122213093587, 25196997894058490607106028095, 24001522306527907199721466108488, 26102037346800387738363882455862531

Offset: 1

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 2.

Row 3 of A369923.

Cf. A190826, A348814, A292409; A003436, A348815, A348818, A348821.

a(14) onwards from Andrew Howroyd, Feb 05 2024

A348818 a(n) = number of chord labeled loopless diagrams by number of K_5.

Original entry on oeis.org

0, 1, 866, 4446741, 55279816356, 1450728060971387, 72078730629785795963, 6235048155225093080061949, 879601407931825739964190440635, 192100729970218737700046212217095291, 62258393664270652226502315136978421947948, 28913744296806659870889046765907226809528931041

Offset: 1

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 4.

Row 5 of A369923.

Cf. A190833, A348819, A348820; A003436, A348813, A348815, A348821.

a(9) onwards from Andrew Howroyd, Feb 05 2024

A348822 a(n) = number of loopless diagrams by number of K_6 up to rotational symmetry.

Original entry on oeis.org

0, 1, 335, 11508322, 1324603148183, 404760320241653655, 282780723811372935744420

Offset: 1

Michael De Vlieger, Nov 01 2021

Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 5.

Cf. A190835, A348821, A348823; A007474, A348814, A348816, A348819.

A348823 a(n) = number of loopless diagrams by number of K_6 up to all symmetries.

Original entry on oeis.org

0, 1, 203, 5765385, 662305416760, 202380163158922023, 141390361908351519807928

Offset: 1

Michael De Vlieger, Nov 01 2021

Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 5.

Cf. A190835, A348821, A348822; A003437, A292409, A348817, A348820.

cp's OEIS Frontend

A348815 a(n) = number of chord labeled loopless diagrams by number of K_4.

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A369923 Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.

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A348813 a(n) = number of chord labeled loopless diagrams by number of K_3.

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A348818 a(n) = number of chord labeled loopless diagrams by number of K_5.

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A348822 a(n) = number of loopless diagrams by number of K_6 up to rotational symmetry.

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A348823 a(n) = number of loopless diagrams by number of K_6 up to all symmetries.

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