A321670 -id:A321670 - 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

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

A322146 Number of permutations of 10 copies of 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 2, 124948668, 174702663548149248, 2360719028641481267959955040, 211490077066069537208795610578715159120, 94446414948214202156311984061437135600678877848560

Offset: 0

Seiichi Manyama, Nov 28 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..59

Row 10 of A322093.

Cf. A321670, A322127, A322128.

a(n) = n! * A321670(n).

a(n) = Integral_{0..infinity} (Sum_{k=1..10} (-1)^(10-k) * binomial(9, 10-k) * x^k/k!)^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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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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Formula

A322146 Number of permutations of 10 copies of 1..n with no element equal to another within a distance of 1.

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Author

Keywords

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Formula