A366540 - OEIS

%I #14 Nov 18 2023 17:51:24

%S 1,6,77,2100,122694,15459444,4220448492,2504511958416,

%T 3237984252258520,9135209892155631900,56307395363835925220625,

%U 758909821892432106622050000,22380517337924566377547475850000,1444814401119843300312851440909560000,204254167807759030850057485271236254820800

%N Number of commutative binary operators defined on the finite chain L_n={0,1,...n}, C:L_n^2-> L_n, which are increasing in each argument, and satisfy the boundary conditions C(0,n)=C(n,0)=0 and C(n,n)=n.

%C The number of commutative discrete conjunctions defined on the finite chain L_n={0,1,...n}, i.e., the number of monotonic increasing binary functions C:L_n^2->L_n such that C(0,n)=C(n,0)=0 and C(n,n)=n (discrete conjunctions), and C(x,y)=C(y,x) for all x,y in L_n (commutative).

%C Also, the number of commutative discrete disjunctions defined on the finite chain L_n={0,1,...n}, i.e., the number of monotonic increasing binary functions D:L_n^2->L_n such that D(0,n)=C(n,0)=n and C(0,0)=0 (discrete disjunctions), and D(x,y)=D(y,x) for all x,y in L_n (commutative).

%C Also, the number of discrete implications defined on the finite chain L_n={0,1,...n} satisfying the contrapositive symmetry with respect to the unary operator N(x)=n-x, for all x in L_n, i.e., the number of binary functions I:L_n^2->L_n such that I is decreasing in the first argument, increasing in the second argument, I(0,0)=I(n,n)=0 and I(n,0)=0 (discrete implication), and I(x,y)=I(N(y),N(x)) for all x,y in L_n (contrapositive symmetry).

%H Marc Munar, S. Massanet and D. Ruiz-Aguilera, <a href="https://doi.org/10.1016/j.ins.2022.10.121">On the cardinality of some families of discrete connectives</a>, Information Sciences, Volume 621, 2023, 708-728.

%F a(n) = Product_{i=1..n} ((2i+n-1)/(2i-1) * Product_{j=i+1..n} (i+j+n-1)/(i+j-1)) - Product_{i=1..n} ((2i+n-2)/(2i-1) * Product_{j=i+1..n} (i+j+n-2)/(i+j-1)).

%F From _Vaclav Kotesovec_, Nov 18 2023: (Start)

%F a(n) = BarnesG(n)^(3/2) * sqrt(BarnesG(3*n)) * Gamma(n)^(3/2) * (Gamma(n/2) * Gamma(3*n) - 2^n*Gamma(3*n/2) * Gamma(2*n)) / (BarnesG(2*n)^(3/2) * sqrt(2*Gamma(n/2) * Gamma(3*n/2)) * Gamma(2*n)^2).

%F a(n) ~ exp(1/24) * 3^(9*n^2/4 + 3*n/4 - 1/24) / (sqrt(A) * n^(1/24) * 2^(3*n^2 + n/2 + 1/8)), where A is the Glaisher-Kinkelin constant A074962. (End)

%t Table[Product[(2*i + n - 1)/(2*i - 1)*

%t     Product[(i + j + n - 1)/(i + j - 1), {j, i + 1, n}], {i, 1, n}] -

%t   Product[(2*i + n - 2)/(2*i - 1)*

%t     Product[(i + j + n - 2)/(i + j - 1), {j, i + 1, n}], {i, 1,

%t     n}], {n, 1, 19}]

%t Table[BarnesG[n]^(3/2) * Sqrt[BarnesG[3*n]] * Gamma[n]^(3/2) * (Gamma[n/2] * Gamma[3*n] - 2^n*Gamma[3*n/2] * Gamma[2*n]) / (BarnesG[2*n]^(3/2) * Sqrt[2*Gamma[n/2] * Gamma[3*n/2]] * Gamma[2*n]^2), {n,1,20}] (* _Vaclav Kotesovec_, Nov 18 2023 *)

%Y Commutative counterpart of operators enumerated in A360612.

%K nonn

%O 1,2

%A _Marc Munar_, Oct 12 2023