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.

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 24 2018

Status

approved

A308356 A(n,k) = (1/k!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

1, 0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, -1, 1, 0, 1, 5, 5, 0, 1, 0, -1, 36, -120, 15, -1, 1, 0, 1, 329, 6286, 2380, 56, 0, 1, 0, -1, 3655, -557991, 1056496, -52556, 203, -1, 1, 0, 1, 47844, 74741031, 1006985994, 197741887, 1192625, 757, 0, 1

Offset

0,18

Links

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Formula

A(n,k) = Sum_{i=k..k*n} b(i) where Sum_{i=k..k*n} b(i) * (-x)^i/i! = (1/k!) * (Sum_{i=1..n} x^i/i!)^k.

Example

For (n,k) = (3,2), (1/2) * (Sum_{i=1..3} x^i/i!)^2 = (1/2) * (x + x^2/2 + x^3/6)^2 = (-x)^2/2 + (-3)*(-x)^3/6 + 7*(-x)^4/24 + (-10)*(-x)^5/120 + 10*(-x)^6/720. So A(3,2) = 1 - 3 + 7 - 10 + 10 = 5.
Square array begins:
   1,  0,   0,       0,           0,                0, ...
   1, -1,   1,      -1,           1,               -1, ...
   1,  0,   1,       5,          36,              329, ...
   1, -1,   5,    -120,        6286,          -557991, ...
   1,  0,  15,    2380,     1056496,       1006985994, ...
   1, -1,  56,  -52556,   197741887,   -2063348839223, ...
   1,  0, 203, 1192625, 38987482590, 4546553764660831, ...

Crossrefs

Columns k=0..4 give A000012, (-1)*A000035, A307349, (-1)*A307350, A307351.

Rows n=0..5 give A000007, A033999, A278990, A308363, A308389, A308390.

Main diagonal gives A308327.

Cf. A144510.

Keyword

sign,tabl

Author

Seiichi Manyama, May 21 2019

Status

approved