A190837 -id:A190837 - 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

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

A322096 Number of permutations of 8 copies of 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 2, 2403588, 36734931452736, 3470403228952634903280, 1490944857678655357195402606800, 2315418264816304038508896461231618573280, 10937192762438008527903830198163831816546577931520

Offset: 0

Seiichi Manyama, Nov 28 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..71

Row 8 of A322093.

Cf. A190837, A322127, A322128.

a(n) = n! * A190837(n).

a(n) = Integral_{0..infinity} (Sum_{k=1..8} (-1)^(8-k) * binomial(7, 8-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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A322096 Number of permutations of 8 copies of 1..n with no element equal to another within a distance of 1.

Views

Author

Keywords

Links

Crossrefs

Formula