A370758 Number of ramified partitions (I,J) of size n, where J is balanced with respect to up brackets and down brackets.

1, 1, 5, 48, 747, 17040, 531810, 21634515, 1107593235, 69482175840, 5229801016650, 464302838867175, 47939015445032250, 5688437019459319125, 767922887039461928775, 116915022542869964287875, 19922514312608630279431875, 3774243527942494591068084000, 790220453914362566924533955250

Offset: 0

Author

Diego Arcis, Feb 29 2024

Keywords

• nonn

Comments

a(n) is the cardinality of the balanced ramified Brauer monoid bBr_n.

Examples

			a(3) = 48 is the number of ramified partitions (I,J) of size 3, in which each block of J contains the same number of up brackets and down brackets from I, i.e., each block of J contains either no brackets from I or one up and one down bracket from I.
		

Links

• Diego Arcis, Table of n, a(n) for n = 0..200
• Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Brauer and Jones tied monoids, arXiv:2107.04170 [math.RT], 2021.
• Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Brauer and Jones tied monoids, J. Pure. Appl. Algebra 227 (2023), 107161.

Crossrefs

Cf. A343254.

Formula

a(n) = Sum_{k=0..n/2} n!^2/(2^(2*k)*k!^2*(n-2*k)!) * A343254(n,k).