A059025 Triangle of Stirling numbers of order 6.

1, 1, 1, 1, 1, 1, 1, 462, 1, 1716, 1, 4719, 1, 11440, 1, 25883, 1, 56134, 1, 118456, 2858856, 1, 245480, 23279256, 1, 502588, 124710300, 1, 1020680, 551496660, 1, 2061709, 2181183147, 1, 4149752, 8021782197, 1, 8333153, 28051272535

Offset: 6

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 6. This is the 6-associated Stirling number of the second kind.

This is entered as a triangular array. The entries S_6(n,k) are zero for 6k>n, so these values are omitted. Initial entry in sequence is S_6(6,1).

Rows are of lengths 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, ...

Examples

			There are 462 ways of partitioning a set N of cardinality 12 into 2 blocks each of cardinality at least 6, so S_6(12,2)=462.
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

Links

• Michael De Vlieger, Table of n, a(n) for n = 6..13205 (rows n = 6..400, flattened).
• Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 5.
• A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Crossrefs

Cf. A008299, A059022, A059023, A059024.

Programs

Mathematica

S6[n_ /; 6 <= n <= 11, 1] = 1; S6[n_, k_] /; 1 <= k <= Floor[n/6] := S6[n, k] = k*S6[n-1, k] + Binomial[n-1, 5]*S6[n-6, k-1]; S6[, ] = 0; Flatten[ Table[ S6[n, k], {n, 6, 24}, {k, 1, Floor[n/6]}]] (* Jean-François Alcover, Feb 21 2012 *)

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=6.

G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t - Sum_{i=0..r-1} t^i/i!)).