A190833 -id:A190833 - cp's OEIS frontend

A322013 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k introduced in order 1..k with no element equal to another within a distance of 1.

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 36, 29, 1, 0, 1, 329, 1721, 182, 1, 0, 1, 3655, 163386, 94376, 1198, 1, 0, 1, 47844, 22831355, 98371884, 5609649, 8142, 1, 0, 1, 721315, 4420321081, 182502973885, 66218360625, 351574834, 56620, 1, 0

Offset: 1

Seiichi Manyama, Nov 24 2018

			Square array begins:
   1, 1,     1,         1,              1,                    1, ...
   0, 1,     5,        36,            329,                 3655, ...
   0, 1,    29,      1721,         163386,             22831355, ...
   0, 1,   182,     94376,       98371884,         182502973885, ...
   0, 1,  1198,   5609649,    66218360625,     1681287695542855, ...
   0, 1,  8142, 351574834, 47940557125969, 16985819072511102549, ...

Seiichi Manyama, Antidiagonals n = 1..53, flattened
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.

Columns k=2..10 give A000012, A190917, A190918, A190920, A190923, A190927, A190932, A321987, A322061.
Rows n=1..10 give A000012, A278990, A190826, A190830, A190833, A190835, A190836, A190837, A321669, A321670.
Main diagonal gives A321666.
Cf. A322093, A369923.

q(n,x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n,k) = subst(serlaplace(q(n,x)^k), x, 1)/k! \\ Andrew Howroyd, Feb 03 2024

T(n,k) = A322093(n,k) / k!. - Andrew Howroyd, Feb 03 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

A348819 a(n) = number of loopless diagrams by number of K_5 up to rotational symmetry.

Original entry on oeis.org

0, 1, 60, 222477, 2211192688, 48357603758012, 2059392303708166507, 155876203880714141444480

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 4.

Cf. A190833, A348818, A348820; A007474, A348814, A348816, A348822.

A321666 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms and introduced in ascending order.

Original entry on oeis.org

1, 1, 1, 29, 94376, 66218360625, 16985819072511102549, 2421032324142610480402567434373, 271259741131895052775392614041761701799270286, 32119646666355552112999645991677870426882424139287301894021793

Offset: 0

Seiichi Manyama, Nov 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..27

Main diagonal of A322013.

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A278990, A321634.

{a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, (-1)^(n-j)*binomial(n-1, j-1)*x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 27 2019

a(n) = A321634(n)/n!.

a(n) ~ exp(5/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 24 2018

A321669 Number of permutations of 9 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 2872754, 104650147201049, 23575497690601916022516, 24302858067615766089801166488125, 91155245844064069307740171414201519055298, 1046031892354833895113128900608177633584652958677057, 32119646666355552112999645991677870426882424139287301894021793

Offset: 0

Seiichi Manyama, Nov 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..70

Row n=9 of A322013.

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A321670.

a(n) ~ 9^(7*n + 1/2) * n^(8*n) / (4480^n * exp(8*(n+1))). - Vaclav Kotesovec, Nov 24 2018

A321670 Number of permutations of 10 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 20824778, 7279277647839552, 19672658572012343899666292, 293736218147318801678882792470437721, 18739368045280595665934917472507368174737872589, 4204427313459831775866154680419213479057724331798640498651

Offset: 0

Seiichi Manyama, Nov 16 2018

nonn

In general, for r > 1, row r of A322013 is asymptotic to r^(r*n + 1/2) * n^((r-1)*n) / ((r!)^n * exp((r-1)*(n+1))). - Vaclav Kotesovec, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 0..63

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A321669.

Row r=10 of A322013.

a(n) ~ 2^(2*n + 1/2) * 5^(8*n + 1/2) * n^(9*n) / (567^n * exp(9*(n+1))). - Vaclav Kotesovec, Nov 24 2018

A348820 a(n) = number of loopless diagrams by number of K_5 up to all symmetries.

Original entry on oeis.org

0, 1, 42, 112418, 1105696796, 24178822553773, 1029696155560021174, 77938101941693076258854

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 4.

Cf. A190833, A348818, A348819; A003437, A292409, A348817, A348823.

A322126 Number of permutations of the multiset {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,...,n,n,n,n,n} with no two consecutive terms equal.

Original entry on oeis.org

1, 0, 2, 7188, 134631576, 7946203275000, 1210527140790855600, 411490045733601418421040, 280031356887267221923677137280, 351026687723982522494728236869341440, 758933713536173718404757245269681913222400

Offset: 0

Seiichi Manyama, Nov 27 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..106

Row 5 of A322093.

Cf. A190833, A322127, A322128.

a[n_] := Integrate[(x - 2 * x^2 + x^3 - 1/6 * x^4 + 1/120 * x^5)^n * Exp[-x],  {x, 0, Infinity}]; Array[a, 10, 0] (* Stefano Spezia, Nov 27 2018 *)

a(n) = n! * A190833(n).

a(n) = Integral_{0..infinity} (x - 2 * x^2 + x^3 - 1/6 * x^4 + 1/120 * x^5)^n * exp(-x) dx.

cp's OEIS Frontend

A322013 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k introduced in order 1..k with no element equal to another within a distance of 1.

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

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A348819 a(n) = number of loopless diagrams by number of K_5 up to rotational symmetry.

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A321666 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms and introduced in ascending order.

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A321669 Number of permutations of 9 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A321670 Number of permutations of 10 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A348820 a(n) = number of loopless diagrams by number of K_5 up to all symmetries.

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A322126 Number of permutations of the multiset {1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,...,n,n,n,n,n} with no two consecutive terms equal.

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