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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).

%I A343254 #36 Nov 11 2021 20:45:35
%S A343254 1,1,2,1,5,2,15,5,3,52,15,8,203,52,25,16,877,203,89,53,4140,877,354,
%T A343254 197,131,21147,4140,1551,810,512,115975,21147,7403,3643,2193,1496,
%U A343254 678570,115975,38154,17759,10201,6697,4213597,678570,210803,93130,51146,32345,22482
%N 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).
%C A343254 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.
%C A343254 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.
%H A343254 Alois P. Heinz, <a href="/A343254/b343254.txt">Rows n = 0..200, flattened</a>
%H A343254 Francesca Aicardi, <a href="https://www.researchgate.net/publication/352438417_BALANCED_PARTITIONS">Balanced partitions</a>, preprint on researchgate, 2021.
%H A343254 Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, <a href="https://arxiv.org/abs/2107.04170">Brauer and Jones tied monoids</a>, arXiv:2107.04170 [math.RT], 2021.
%F A343254 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.
%e A343254 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).
%e A343254 Triangle T(n,k) begins:
%e A343254         1;
%e A343254         1;
%e A343254         2,      1;
%e A343254         5,      2;
%e A343254        15,      5,      3;
%e A343254        52,     15,      8;
%e A343254       203,     52,     25,    16;
%e A343254       877,    203,     89,    53;
%e A343254      4140,    877,    354,   197,   131;
%e A343254     21147,   4140,   1551,   810,   512;
%e A343254    115975,  21147,   7403,  3643,  2193,  1496;
%e A343254    678570, 115975,  38154, 17759, 10201,  6697;
%e A343254   4213597, 678570, 210803, 93130, 51146, 32345, 22482;
%e A343254   ...
%Y A343254 Cf. A000110 (Bell numbers), A023998, A061691 (generalized Stirling numbers), A344775 (row sums).
%K A343254 nonn,tabf
%O A343254 0,3
%A A343254 _Francesca Aicardi_, Jun 04 2021