0,2

For a nonnegative integer n, [n] denotes the totally ordered set ({0,...,n},<=). The lattice [n]*[n] is obtained by first forming a disjoint union of two copies of [n], then identifying their top and bottom elements. So, for example, [2]*[2] is the square lattice (isomorphic to the product [1]x[1]). [0]*[0] is the one-point lattice, and [1]*[1] is isomorphic to [1].

Linus Bao, Christy Hazel, Tia Karkos, Alice Kessler, Austin Nicolas, Kyle Ormsby, Jeremie Park, Cait Schleff, and Scotty Tilton, Transfer systems for rank two elementary abelian groups: characteristic functions and matchstick games, arXiv preprint, arXiv:2310.13835 [math.CO], 2023.

Linus Bao, Christy Hazel, Tia Karkos, Alice Kessler, Austin Nicolas, Kyle Ormsby, Jeremie Park, Cait Schleff, and Scotty Tilton, Transfer systems for rank two elementary abelian groups: characteristic functions and matchstick games, Tunisian Journal of Mathematics, Vol. 7 (2025), No. 1, 167-191.

a(n) = 2*(b(n)^2 + b(n-1)*(b(n+1) - 2*b(n))) for n > 0, where b(n) is the n-th Catalan number (A000108) [Bao et al. (Corollary 4.12)].

For n = 0, [0]*[0] is the one-point lattice, on which there is a unique transfer system.

For n = 1, [1]*[1] is isomorphic to [1], on which there are exactly two transfer systems.

For n = 2, [2]*[2] is isomorphic to the square lattice [1]x[1], on which there are exactly 10 transfer systems.

a:= proc(n) option remember; `if`(n<2, 1+n,

4*(2*n-1)*(2*n-3)*(n^2+n-1)*a(n-1)/((n+1)*(n+2)*(n^2-n-1)))

end:

seq(a(n), n=0..20); # Alois P. Heinz, Jan 23 2026

If[n==0, 1, 2(CatalanNumber[n]^2 + CatalanNumber[n-1](CatalanNumber[n+1]-2*CatalanNumber[n]))]

Cf. A000108.

Sequence in context: A277493 A361448 A376226 * A346417 A377529 A108205

Adjacent sequences: A392807 A392808 A392810 * A392812 A392813 A392814

nonn,easy

Ben Spitz, Jan 23 2026

approved