Francesca Aicardi - cp's OEIS frontend

User: Francesca Aicardi

Francesca Aicardi has authored 3 sequences.

A358397 Number of pairs of partitions (A<=B, that is, A is a refinement of B) of [n] such that A is noncrossing and its nontrivial blocks are of type {a,b} with a <= n and b > n.

1, 1, 3, 9, 37, 157, 811, 4309, 26327, 164947, 1151477, 8224863, 64158567, 511177515, 4386520201, 38389960685, 358214414675, 3404632390971, 34234771676473, 350261221644771, 3768281045014927, 41210302324325919, 471585931164213345, 5480984322433817771, 66388136273738685321

Offset: 0

Francesca Aicardi, Nov 13 2022

nonn

Cf. A000110.

a(n) = Sum_{k=0..m} binomial(m,k)*binomial(m+e,k)*Bell(n-k), with m = floor(n/2), e = n mod 2 and where Bell(n) is the Bell exponential number A000110(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).

Original entry on oeis.org

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

Francesca Aicardi, Jun 04 2021

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.

			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;
  ...

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.

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

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.

A344775 a(n) is the number of 2-balanced partitions of a set of n elements.

Original entry on oeis.org

1, 1, 3, 7, 23, 75, 296, 1222, 5699, 28160, 151857, 867356, 5302073, 34176364, 232932946, 1665341260, 12487204067, 97743060158, 797730561155, 6768022876452, 59606300409007, 543773719267894, 5131560749880622, 50012790651415626, 502782861641973256, 5206962982060933623

Offset: 0

Francesca Aicardi, May 28 2021

nonn

a(n) is calculated as the sum of the numbers b(n,k) (A343254) of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k. The sum runs over all integers k from zero to floor(n/2).

			For n=3, a(3) = b(3,0) + b(3,1). b(3,0) is the number of partitions of a set of three elements (all elements lie in the third subset), i.e., b(3,0) = Bell(3) = 5. b(3,1) is the number of 2-balanced partitions of a set {p,q,r} in which the first and the second subsets, say {p} and {q}, have cardinality 1. There are only two 2-balanced partitions: {{p,q},{r}}, and {{p,q,r}}. So, b(3,1)=2 and a(3)=7.

Alois P. Heinz, Table of n, a(n) for n = 0..576
Francesca Aicardi, Balanced partitions, preprint on researchgate, 2021.

Row sums of A343254.

Cf. A000110 (Bell numbers).

a(n) = Sum_{k=0..floor(n/2)} A343254(n,k).

a(19)-a(25) from Alois P. Heinz, Jun 16 2021

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User: Francesca Aicardi

A358397 Number of pairs of partitions (A<=B, that is, A is a refinement of B) of [n] such that A is noncrossing and its nontrivial blocks are of type {a,b} with a <= n and b > n.

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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).

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A344775 a(n) is the number of 2-balanced partitions of a set of n elements.

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