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%I #26 Apr 08 2021 00:30:05
%S 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,
%T 1,1,35,280,105,420,210,28,1,1,126,280,1260,945,1260,378,36,1,1,126,
%U 2100,6300,945,4725,3150,630,45,1,1,462,5775,15400,17325,10395,17325,6930,990,55,1
%N 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.
%C This is the generalized form of A060540 that allows for a number of groups that does not evenly divide the number of items.
%H Michael De Vlieger, <a href="/A308624/b308624.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)
%H Johan Kok, <a href="https://doi.org/10.30538/psrp-odam2020.0046">Degree affinity number of certain 2-regular graphs</a>, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 77-84.
%H Johan Kok and Joseph Varghese Kureethara, <a href="https://doi.org/10.22049/cco.2020.26895.1158">Stirling number of the fourth kind and lucky partitions of a finite set</a>, Communications in Combinatorics and Optimization (2021) Vol. 6, Issue 2, 211-219.
%F T(n,k) = n!/((n mod k)!*(k-(n mod k))!*ceiling(n/k)!^(n mod k)*floor(n/k)!^(k-(n mod k))).
%F 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).
%e Example terms in T(5,2) = 10:
%e (1,2,3) (4,5)
%e (1,2,4) (3,5)
%e (1,2,5) (3,4)
%e (1,3,4) (2,5)
%e (1,3,5) (2,4)
%e (1,4,5) (2,3)
%e (1,2) (3,4,5)
%e (1,3) (2,4,5)
%e (1,4) (3,4,5)
%e (1,5) (2,3,4)
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 3, 6, 1;
%e 1, 10, 15, 10, 1;
%e 1, 10, 15, 45, 15, 1;
%e 1, 35, 105, 105, 105, 21, 1;
%e 1, 35, 280, 105, 420, 210, 28, 1;
%e 1, 126, 280, 1260, 945, 1260, 378, 36, 1;
%e 1, 126, 2100, 6300, 945, 4725, 3150, 630, 45, 1;
%e 1, 462, 5775, 15400, 17325, 10395, 17325, 6930, 990, 55, 1;
%t 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}]]
%o (PARI) T(n,k) = my(A = n % k, B = k - (A)); n!/(A!*B!*ceil(n/k)!^A*floor(n/k)!^B);
%o tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Jun 28 2019
%Y Cf. A060540.
%K nonn,tabl
%O 1,5
%A _Dillon Lareau_, Jun 11 2019