A059023 Triangle of Stirling numbers of order 4.

1, 1, 1, 1, 1, 35, 1, 126, 1, 336, 1, 792, 1, 1749, 5775, 1, 3718, 45045, 1, 7722, 231231, 1, 15808, 981981, 1, 32071, 3741738, 2627625, 1, 64702, 13307294, 35735700, 1, 130084, 45172842, 300179880, 1, 260984, 148417854, 2002016016, 1, 522937, 476330361

Offset: 4

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

Keywords

• nonn
• tabf
• nice

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,...

Examples

			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.
		

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.

Crossrefs

Row sums give A057837.

Cf. A008299, A059022, A059024, A059025.

Programs

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 *)

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