A343254 Triangle read by rows: T(n,k) is the number of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k, for n >= 0, k = 0..floor(n/2).
1, 1, 2, 1, 5, 2, 15, 5, 3, 52, 15, 8, 203, 52, 25, 16, 877, 203, 89, 53, 4140, 877, 354, 197, 131, 21147, 4140, 1551, 810, 512, 115975, 21147, 7403, 3643, 2193, 1496, 678570, 115975, 38154, 17759, 10201, 6697, 4213597, 678570, 210803, 93130, 51146, 32345, 22482
Offset: 0
Examples
T(4,1) = 5, number of 2-balanced partitions of a set A of 4 elements with 1 element in the first subset and 1 element in the second subset: A={a} U {b} U {c,d}. The five partitions are: ((a,b),(c),(d)), ((a,b),(c,d)), ((a,b,c),(d)), ((a,b,d),(c)), ((a,b,c,d)). Note that if a block contains a, then it must contain b. Thus, T(n,1) = T(n-1,0).
Triangle T(n,k) begins:
1;
1;
2, 1;
5, 2;
15, 5, 3;
52, 15, 8;
203, 52, 25, 16;
877, 203, 89, 53;
4140, 877, 354, 197, 131;
21147, 4140, 1551, 810, 512;
115975, 21147, 7403, 3643, 2193, 1496;
678570, 115975, 38154, 17759, 10201, 6697;
4213597, 678570, 210803, 93130, 51146, 32345, 22482;
...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
- Francesca Aicardi, Balanced partitions, preprint on researchgate, 2021.
- Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Brauer and Jones tied monoids, arXiv:2107.04170 [math.RT], 2021.
Crossrefs
Formula
T(n,k) = Sum_{j=1..n-k} C(n,k,j). C(n,k,j) is defined for n>=2k, j<=n-k, and obtained by the recursion: C(n,k,j) = C(n-1,k,j-1) + j*C(n-1,k,j), with initial conditions C(2k,k,j) = triangle A061691(k,j) of generalized Stirling numbers.
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