OFFSET
5,7
COMMENTS
The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 5. This is the 5-associated Stirling number of the second kind.
This is entered as a triangular array. The entries S_5(n,k) are zero for 5k>n, so these values are omitted. Initial entry in sequence is S_5(5,1).
Rows are of lengths 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,...
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
LINKS
Alois P. Heinz, Rows n = 5..320, flattened
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
FORMULA
S_r(n+1, k) = k*S_r(n, k) + binomial(n, r-1)*S_r(n-r+1, k-1); for this sequence, r=5.
G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t-sum(t^i/i!, i=0..r-1))).
T(n,k) = Sum_{j=0..min(n/4,k)} (-1)^j*n!/(24^j*j!*(n-4j)!)*S_4(n-4j,k-j), where S_4 are the 4-associated Stirling numbers of the second kind A059023. - Fabián Pereyra, Feb 21 2022
EXAMPLE
There are 126 ways of partitioning a set N of cardinality 10 into 2 blocks each of cardinality at least 5, so S_5(10,2) = 126.
Triangle begins:
1;
1;
1;
1;
1;
1, 126;
1, 462;
1, 1254;
1, 3003;
1, 6721;
1, 14443, 126126;
1, 30251, 1009008;
1, 62322, 5309304;
1, 127024, 23075052;
1, 257108, 89791416;
1, 518092, 325355316, 488864376;
...
MAPLE
T:= proc(n, k) option remember; `if`(k<1 or k>n/5, 0,
`if`(k=1, 1, k*T(n-1, k)+binomial(n-1, 4)*T(n-5, k-1)))
end:
seq(seq(T(n, k), k=1..n/5), n=5..25); # Alois P. Heinz, Aug 18 2017
MATHEMATICA
S5[n_ /; 5 <= n <= 9, 1] = 1; S5[n_, k_] /; 1 <= k <= Floor[n/5] := S5[n, k] = k*S5[n-1, k] + Binomial[n-1, 4]*S5[n-5, k-1]; S5[_, _] = 0; Flatten[ Table[ S5[n, k], {n, 5, 25}, {k, 1, Floor[n/5]}]] (* Jean-François Alcover, Feb 21 2012 *)
CROSSREFS
KEYWORD
nonn,tabf,nice
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
STATUS
approved