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

1,8

Links

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.

Formula

T(n,k) = A322093(n,k) / k!. - Andrew Howroyd, Feb 03 2024

Example

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, ...

Prog

(PARI)
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

Crossrefs

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.

Sequence in context: A157012 A102365 A269945 * A102259 A021200 A265301

Adjacent sequences: A322010 A322011 A322012 * A322014 A322015 A322016

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 24 2018

Status

approved