OFFSET
4,6
COMMENTS
The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 4. This is the 4-associated Stirling number of the second kind.
This is entered as a triangular array. The entries S_4(n,k) are zero for 4k>n, so these values are omitted. Initial entry in sequence is S_4(4,1).
Rows are of lengths 1,1,1,1,2,2,2,2,3,3,3,3,...
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 = 4..300, 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=4.
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/3,k)} (-1)^j*n!/(6^j*j!*(n-3j)!)*S_3(n-3j,k-j), where S_3 are the 3-associated Stirling numbers of the second kind A059022. - Fabián Pereyra, Feb 21 2022
EXAMPLE
There are 35 ways of partitioning a set N of cardinality 8 into 2 blocks each of cardinality at least 4, so S_4(8,2) = 35.
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, add(
expand(x*b(n-j))*binomial(n-1, j-1), j=4..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=4..20); # Alois P. Heinz, Feb 21 2022
# alternative
A059023 := proc(n, k)
option remember;
if n<4 then
0;
elif n < 8 and k=1 then
1 ;
else
k*procname(n-1, k)+binomial(n-1, 3)*procname(n-4, k-1) ;
end if;
end proc: # R. J. Mathar, Apr 15 2022
MATHEMATICA
s4[n_, k_] := k*s4[n-1, k] + Binomial[n-1, 3]*s4[n-4, k-1]; s4[n_, k_] /; 4 k > n = 0; s4[_, k_ /; k <= 0] = 0; s4[0, 0] = 1;
Flatten[Table[s4[n, k], {n, 4, 20}, {k, 1, Floor[n/4]}]][[1 ;; 42]] (* Jean-François Alcover, Jun 16 2011 *)
CROSSREFS
KEYWORD
nonn,tabf,nice
AUTHOR
Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
STATUS
approved