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A366540
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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.
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1
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1, 6, 77, 2100, 122694, 15459444, 4220448492, 2504511958416, 3237984252258520, 9135209892155631900, 56307395363835925220625, 758909821892432106622050000, 22380517337924566377547475850000, 1444814401119843300312851440909560000, 204254167807759030850057485271236254820800
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OFFSET
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1,2
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COMMENTS
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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).
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).
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).
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LINKS
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FORMULA
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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)).
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).
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)
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MATHEMATICA
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Table[Product[(2*i + n - 1)/(2*i - 1)*
Product[(i + j + n - 1)/(i + j - 1), {j, i + 1, n}], {i, 1, n}] -
Product[(2*i + n - 2)/(2*i - 1)*
Product[(i + j + n - 2)/(i + j - 1), {j, i + 1, n}], {i, 1,
n}], {n, 1, 19}]
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 *)
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CROSSREFS
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Commutative counterpart of operators enumerated in A360612.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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