A366446 Number of discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n}.

4, 136, 21238, 15374304, 52326659814, 845020424183364, 65102985676317726176, 24005912569370916926920192, 42445533661127789112292364580400, 360256756545210313397342412375573121875, 14686785417300337272290307023148088973414062500

Offset

1,1

Comments

The number of discrete aggregation functions on the finite chain L_n={0,1,...,n-1,n}, i.e., the number of monotonic increasing binary functions F: L_n^2->L_n such that F(0,0)=0 and F(n,n)=n.

Links

Table of n, a(n) for n=1..11.

M. Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, Volume 621, 2023, 708-728.

Formula

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

From Vaclav Kotesovec, Nov 18 2023: (Start)

a(n) = BarnesG(n+1)^3 * BarnesG(3*n+1) * (3*Gamma(n) * Gamma(3*n) / Gamma(2*n)^2 - 2) * (3*(3*n+1) * Gamma(n) * Gamma(3*n) / Gamma(2*n)^2 - 2*(n-1)) / (16*(2*n+1) * BarnesG(2*n+1)^3)).

a(n) ~ exp(1/12) * 3^(9*n^2/2 + 6*n + 23/12) / (A * n^(1/12) * 2^(6*n^2 + 8*n + 11/4)), where A is the Glaisher-Kinkelin constant A074962. (End)

Mathematica

Table[Product[
   Product[Product[(i + j + k - 1)/(i + j + k - 2), {k, 1, n}], {j, 1,
      n + 1}], {i, 1, n + 1}] -
  2*Product[
    Product[Product[(i + j + k - 1)/(
      i + j + k - 2), {k, 1, n - 1}], {j, 1, n + 1}], {i, 1, n + 1}] +
   Product[Product[
    Product[(i + j + k - 1)/(i + j + k - 2), {k, 1, n - 2}], {j, 1,
     n + 1}], {i, 1, n + 1}], {n, 2, 13}]
Table[BarnesG[n + 1]^3 * BarnesG[3*n + 1]*(3*Gamma[n] * Gamma[3*n]/Gamma[2*n]^2 - 2) * (3*(3*n + 1)*Gamma[n]*Gamma[3*n]/Gamma[2*n]^2 - 2*(n-1)) / (16*(2*n + 1) * BarnesG[2*n + 1]^3), {n, 1, 13}] (* Vaclav Kotesovec, Nov 18 2023 *)

Crossrefs

Sequence in context: A012052 A054052 A012070 * A001374 A229416 A155207

Adjacent sequences: A366443 A366444 A366445 * A366447 A366448 A366449

Keyword

nonn

Author

Marc Munar, Oct 10 2023

Status

approved