A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.

1, 1, 1, 1, 3, 1, 0, 7, 6, 1, 0, 10, 25, 10, 1, 0, 10, 75, 65, 15, 1, 0, 0, 175, 315, 140, 21, 1, 0, 0, 280, 1225, 980, 266, 28, 1, 0, 0, 280, 3780, 5565, 2520, 462, 36, 1, 0, 0, 0, 9100, 26145, 19425, 5670, 750, 45, 1, 0, 0, 0, 15400, 102025, 125895, 56595, 11550, 1155

Offset

1,5

Comments

a(n,k) = 0 if k > n; a(n,k) = 0 if n > 0 and k < 0; a(n,k) can be extended to negative n and k, just as the Stirling numbers or Pascal's triangle can be extended. The present triangle is called the tri-restricted Stirling numbers of the second kind.

Also the Bell transform of the sequence "a(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

References

J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.

Links

Table of n, a(n) for n=1..64.

J. Y. Choi and J. D. H. Smith, The Tri-restricted Numbers and Powers of Permutation Representations, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.

J. Y. Choi and J. D. H. Smith, On the Unimodality and Combinatorics of the Bessel Numbers, Discrete Math., 264 (2003), 45-53.

J. Y. Choi et al., Reciprocity for multirestricted Stirling numbers, J. Combin. Theory 113 A (2006), 1050-1060.

Formula

a(n, k) = a(n-1, k-1) + k*a(n-1, k) - binomial(n-1, 3)*a(n-4, k-1).

G.f. = Sum_{k_1+k_2+k_3=k, k_1+ 2k_2+3k_3=n} frac{n!}{(1!)^{k_1}(2!)^{k_2}(3!)^{k_3}k_1!k_2!k_3!}.

E.g.f.: exp(y*(x+x^2/2+x^3/6)). - Vladeta Jovovic, Nov 01 2005

Example

a(1,1)=1;
a(2,1)=1; a(2,2)=1;
a(3,1)=1; a(3,2)=3; a(3,3)=1;
a(4,1)=0; a(4,2)=7; a(4,3)=6; a(4,4)=1;
a(5,1)=0; a(5,2)=10; a(5,3)=25; a(5,4)=10; a(5,5)=1;
a(6,1)=0; a(6,2)=10; a(6,3)=75; a(6,4)=65; a(6,5)=15; a(6,6)=1; ...

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> `if`(n<3, 1, 0), 10); # Peter Luschny, Jan 27 2016

Mathematica

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# < 3, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Prog

(PARI) row(n) = {x='x+O('x^(n+1)); polcoeff(serlaplace(exp(y*(x+x^2/2+x^3/6))), n, 'x); }
tabl(nn) = for(n=1, nn, print(Vecrev(row(n)/y))) \\ Jinyuan Wang, Dec 21 2019

Crossrefs

A144385 and A144402 are other versions of this same triangle.

Cf. A001680, A008277 (stirling numbers).

Sequence in context: A176108 A110504 A361475 * A206306 A178124 A354010

Adjacent sequences: A111243 A111244 A111245 * A111247 A111248 A111249

Keyword

nonn,tabl

Author

Ji Young Choi, Oct 31 2005

Extensions

More terms from Vladeta Jovovic, Nov 01 2005

Recurrence, offset and example corrected by David Applegate, Jan 16 2009

Status

approved