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

0,3

Comments

A 2-balanced partition is a partition of a set which is the union of three subsets, with the property that the cardinality of the first two subsets are equal (possibly zero), and each block contains the same number (possibly zero) of elements from the first and from the second subset. The rows add to A344775.

T(n,0) are the Bell numbers. T(2k,k) are the numbers of 2-balanced partitions in the particular case in which the third set is empty. T(2k,k) are the generalized Bell numbers given in A023998.

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.

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.

Example

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;
  ...

Crossrefs

Cf. A000110 (Bell numbers), A023998, A061691 (generalized Stirling numbers), A344775 (row sums).

Sequence in context: A192263 A109738 A258307 * A143891 A030400 A115345

Adjacent sequences: A343251 A343252 A343253 * A343255 A343256 A343257

Keyword

nonn,tabf

Author

Francesca Aicardi, Jun 04 2021

Status

approved