A369923 Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.

0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 31, 22, 1, 0, 1, 293, 1415, 134, 1, 0, 1, 3326, 140343, 75843, 866, 1, 0, 1, 44189, 20167651, 83002866, 4446741, 5812, 1, 0, 1, 673471, 3980871156, 158861646466, 55279816356, 276154969, 40048, 1, 0

Offset

1,8

Comments

Also, T(n,k) is the number of generalized chord labeled loopless diagrams with k parts of K_n. See the Krasko reference for a full definition.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 51 antidiagonals)

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.

Example

Array begins:
n\k| 1 2    3         4              5                    6 ...
---+-----------------------------------------------------------
 1 | 0 1    1         1              1                    1 ...
 2 | 0 1    4        31            293                 3326 ...
 3 | 0 1   22      1415         140343             20167651 ...
 4 | 0 1  134     75843       83002866         158861646466 ...
 5 | 0 1  866   4446741    55279816356     1450728060971387 ...
 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ...
 ...

Mathematica

T[n_, k_] := If[k == 1, 0, Expand[(-1)^(k (n + 1))/(k - 1)! n Hypergeometric1F1[1 - n, 2, x]^k x^(k - 1)] /. x^p_ :> p!] (* Eric W. Weisstein, Feb 20 2025 *)

Prog

(PARI) \\ compare with A322013.
q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0)

Crossrefs

Rows 2..6 are A003436, A348813, A348815, A348818, A348821.

Column 3 is A197657, column 4 appears to be A209183(n)/2.

Cf. A322013 (without linearly adjacent elements equal), A322093.

Sequence in context: A281891 A124539 A351703 * A249094 A096501 A062862

Adjacent sequences: A369920 A369921 A369922 * A369924 A369925 A369926

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 05 2024

Status

approved