A308624 Triangle read by rows: T(n,k) = n!/((n mod k)!*(k-(n mod k))!*(ceiling(n/k)!^(n mod k))*(floor(n/k)!^(k-(n mod k)))), n >= 1, n >= k >= 1, the number of ways of dividing n labeled items into k unlabeled boxes as evenly as possible.

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

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

Cf. A060540.

Sequence in context: A113046 A245541 A209563 * A133825 A365968 A156710

Adjacent sequences: A308621 A308622 A308623 * A308625 A308626 A308627

Keyword

nonn,tabl

Author

Dillon Lareau, Jun 11 2019

Status

approved