OFFSET
1,5
COMMENTS
This is the generalized form of A060540 that allows for a number of groups that does not evenly divide the number of items.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Johan Kok, Degree affinity number of certain 2-regular graphs, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 77-84.
Johan Kok and Joseph Varghese Kureethara, Stirling number of the fourth kind and lucky partitions of a finite set, Communications in Combinatorics and Optimization (2021) Vol. 6, Issue 2, 211-219.
FORMULA
T(n,k) = n!/((n mod k)!*(k-(n mod k))!*ceiling(n/k)!^(n mod k)*floor(n/k)!^(k-(n mod k))).
T(n,k) = n!/(A!*B!*ceiling(n/k)!^A*floor(n/k)!^B) where A = n mod k and B = k - (n mod k).
EXAMPLE
Example terms in T(5,2) = 10:
(1,2,3) (4,5)
(1,2,4) (3,5)
(1,2,5) (3,4)
(1,3,4) (2,5)
(1,3,5) (2,4)
(1,4,5) (2,3)
(1,2) (3,4,5)
(1,3) (2,4,5)
(1,4) (3,4,5)
(1,5) (2,3,4)
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 3, 6, 1;
1, 10, 15, 10, 1;
1, 10, 15, 45, 15, 1;
1, 35, 105, 105, 105, 21, 1;
1, 35, 280, 105, 420, 210, 28, 1;
1, 126, 280, 1260, 945, 1260, 378, 36, 1;
1, 126, 2100, 6300, 945, 4725, 3150, 630, 45, 1;
1, 462, 5775, 15400, 17325, 10395, 17325, 6930, 990, 55, 1;
MATHEMATICA
Flatten[Table[n!/(Mod[n, k]!*(k - Mod[n, k])!*Ceiling[n/k]!^Mod[n, k]* Floor[n/k]!^(k - Mod[n, k])), {n, 1, 10}, {k, 1, n}]]
PROG
(PARI) T(n, k) = my(A = n % k, B = k - (A)); n!/(A!*B!*ceil(n/k)!^A*floor(n/k)!^B);
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jun 28 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Dillon Lareau, Jun 11 2019
STATUS
approved