

A039810


Matrix square of Stirling2 Triangle A008277: 2levels set partitions of [n] into k firstlevel subsets.


17



1, 2, 1, 5, 6, 1, 15, 32, 12, 1, 52, 175, 110, 20, 1, 203, 1012, 945, 280, 30, 1, 877, 6230, 8092, 3465, 595, 42, 1, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1, 115975, 2090424, 5971350
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

This triangle groups certain generalized Stirling numbers of the second kind A000558, A000559, ... They can also be interpreted in terms of trees of height 3 with n leaves and constraints on the order of the root.
From Peter Bala, Jul 19 2014: (Start)
The (n,k)th entry in this table gives the number of double partitions of the set [n] = {1,2,...,n} into k blocks. To form a double partition of [n] we first write [n] as a disjoint union X_1 U...U X_k of k nonempty subsets (blocks) X_i of [n]. Then each block X_i is further partitioned into subblocks to give a double partition. For instance, {1,2,4} U {3,5} is a partition of [5] into 2 blocks and {{1,4},{2}} U {{3},{5}} is a refinement of this partition to a double partition of [5] into 2 blocks (and 4 subblocks).
Compare the above interpretation for the (n,k)th entry of this table with the interpretation of the (n,k)th entry of A013609 (the square of Pascal's triangle but with the rows read in reverse order) as counting the pairs (X,Y) of subsets of [n] such that Y = k and X is contained in Y. (End)
Also the Bell transform of the shifted Bell numbers B(n+1) without column 0. For the definition of the Bell transform see A264428.  Peter Luschny, Jan 28 2016
T(n,k) is the number of partitions of an nset into colored blocks, such that exactly k colors are used and the colors are introduced in increasing order. T(3,2) = 6: 1a23b, 13a2b, 12a3b, 1a2a3b, 1a2b3a, 1a2b3b.  Alois P. Heinz, Aug 27 2019


LINKS

Tilman Piesk, First 100 rows, flattened
A. Aboud, J.P. Bultel, A. Chouria, J.G. Luque, O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 20142015.
T. Mansour, A. Munagi, M. Shattuck, Recurrence Relations and TwoDimensional Set Partitions , J. Int. Seq. 14 (2011) # 11.4.1


FORMULA

S2 = A008277 (Stirling numbers of the second kind).
T = (S2)^2.
T(n,k) = Sum_{i=k..n} S2(n,i) * S2(i,k).
E.g.f. kth column: (exp(exp(x)1))^k/k!.
From Peter Bala, Jul 19 2014: (Start)
T(n,k) = Sum_{disjoint unions X_1 U...U X_k = [n]} Bell(X_1)*...*Bell(X_k), where Bell(n) = A000110(n).
Recurrence equation: T(n+1,k+1) = Sum_{j = k..n} Bell(n+1j)*binomial(n,j)* T(j,k).
Row sums [1,3,12,60,358,...] = A000258. (End)


EXAMPLE

Triangle begins:
k = 1 2 3 4 5 sum
n
1 1 1
2 2 1 3
3 5 6 1 12
4 15 32 12 1 60
5 52 175 110 20 1 358
Matrix multiplication Stirling2 * Stirling2:
1 0 0 0
1 1 0 0
1 3 1 0
1 7 6 1
.
1 0 0 0 1 0 0 0
1 1 0 0 2 1 0 0
1 3 1 0 5 6 1 0
1 7 6 1 15 32 12 1
From Peter Bala, Jul 19 2014: (Start)
T(5,2) = 175: A 5set can be partitioned into 2 blocks as either a union of a 3set and a 2set or as a union of a 4set and a singleton set.
In the first case there are 10 ways of partitioning a 5set into a 3set and a 2set. Each 3set can be further partitioned into subblocks in Bell(3) = 5 ways and each 2set can be further partitioned into subblocks in Bell(2) = 2 ways. So altogether we obtain 10*5*2 = 100 double partitions of this type.
In the second case, there are 5 ways of partitioning a 5set into a 4set and a 1set. Each 4set can be further partitioned in Bell(4) = 15 ways and each 1set can be further partitioned in Bell(1) = 1 way. So altogether we obtain 5*15*1 = 75 double partitions of this type.
Hence, in total, T(5,2) = 100 + 75 = 175. (End)


MAPLE

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n > combinat:bell(n+1), 10); # Peter Luschny, Jan 28 2016


MATHEMATICA

Flatten[Table[Sum[StirlingS2[n, i]*StirlingS2[i, k], {i, k, n}], {n, 1, 10}, {k, 1, n}]] (* Indranil Ghosh, Feb 22 2017 *)
rows = 10;
t = Table[BellB[n+1], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* JeanFrançois Alcover, Jun 22 2018, after Peter Luschny *)


CROSSREFS

Cf. A039811, A039814, A039813 (other products of Stirling matrices).
T(n, 1) = A000110(n) (first column) (Bell numbers).
T(n, 2) = A000558(n) 2levels set partitions with 2 firstlevel classes.
T(n, n1) = A002378(n1) = n*(n1) = 2*C(n,2) = setpartitions into (n2) singletons and one of the two possible set partitions of [2].
Sum is A000258(n), 2levels set partitions.
Another version with offset 0: A130191.
Horizontal mirror triangle is A046817.
T(2n,n) gives A321712.
Sequence in context: A128567 A217204 A179455 * A328297 A124575 A178121
Adjacent sequences: A039807 A039808 A039809 * A039811 A039812 A039813


KEYWORD

nonn,tabl


AUTHOR

Christian G. Bower, Feb 15 1999


EXTENSIONS

Definition and interpretation edited by Olivier Gérard, Jul 31 2011


STATUS

approved



