A360612 -id:A360612 - cp's OEIS frontend

A366540 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.

1, 6, 77, 2100, 122694, 15459444, 4220448492, 2504511958416, 3237984252258520, 9135209892155631900, 56307395363835925220625, 758909821892432106622050000, 22380517337924566377547475850000, 1444814401119843300312851440909560000, 204254167807759030850057485271236254820800

Offset: 1

Marc Munar, Oct 12 2023

nonn

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).

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

Commutative counterpart of operators enumerated in A360612.

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 *)

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)).
From Vaclav Kotesovec, Nov 18 2023: (Start)
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)

A367192 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the left neutrality principle, i.e., I(n,y)=y for all y in L_n.

Original entry on oeis.org

1, 5, 84, 4719, 884884, 553361016, 1153471856900, 8012241391384695, 185424118272842096128, 461964068878932837522210816

Offset: 1

Marc Munar, Nov 09 2023

Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the left neutrality principle, 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)=n and I(n,0)=0 (discrete implication), and I(n,y)=y for all y in L_n (left neutrality principle).

The proposed formula is recursive and implemented using dynamic programming using Python. and only the first 10 terms could be obtained. See github link.

Marc Munar, Python program.
Marc Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, Volume 621, 2023, 708-728.
Marc Munar, S. Massanet and D. Ruiz-Aguilera, DiscreteFuzzyOperators - A Python library for computing with fuzzy operators, Zenodo, Version 1.13.

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

Python
```
See Github link
```

a(n)=G((1,2,...,n)), where G(v) is defined recursively as:
   ·G(v)=det(A(v))-Sum_{x in V_n(v)\v} G(v), where:
       · A(v)_{i,j}=binomial(n+v_j, n-i+j).
       · V_n(v) is the set of decreasing vectors x of n components, whose entries are taken from L_n, and x_i<=v_i for all i in {1,...,n}.
   ·G(v)=Binomial(n+x-1,x), if v=(x,0,...,0), with v being a vector of n components and 1<=x<=n.

A367434 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the ordering principle, i.e., I(x,y)=n if and only if x<=y, with x,y in L_n.

Original entry on oeis.org

1, 4, 70, 4125, 803088, 516241856, 1097667526748, 7733677937945025, 180801869068010677500, 14039958885548006432007000, 3624052564790121589865007580800, 3111116716392548007169585391982775200, 8885709622433787461657425956670165208100240, 84456895620767739860514483200734788327292908270424

Offset: 1

Marc Munar, Nov 18 2023

Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the ordering principle, 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)=n and I(n,0)=0 (discrete implication), and I(x,y)=n if and only if x<=y, with x,y in L_n. (ordering principle).

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

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

Table[(Product[Factorial[2 n - i]/Factorial[n - 2 + i], {i, 1, n}]*Product[Product[(3 n + 1 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}] - Product[Factorial[2 n - i - 1]/Factorial[n - 3 + i], {i, 1, n}]*Product[Product[(3 n - 1 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}])*2^((n*(n - 1))/2)*Product[i^(n - i), {i, 1, n - 1}]*Product[1/Factorial[2 n + 1 - 2 i], {i, 1, n}], {n, 1, 20}]
Table[BarnesG[n] * Sqrt[BarnesG[4*n]] * Gamma[n] * (1 - Gamma[3*n - 2]*Gamma[3*n - 1]/(Gamma[2*n - 2]*Gamma[4*n - 1])) / (BarnesG[3*n] * Sqrt[2*Gamma[2*n]]), {n, 1, 20}] (* Vaclav Kotesovec, Nov 18 2023 *)

a(n) = ((Product_{i=1..n} (2n-i)!/(n-2+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n+1-i+2j)/2)-(Product_{i=1..n} (2n-i-1)!/(n-3+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n-1-i+2j)/2))*(2^(n*(n-1)/2))*(Product_{i=1..n} i^(n-i)/(2n+1-2i)!).
From Vaclav Kotesovec, Nov 18 2023: (Start)
a(n) = BarnesG(n) * sqrt(BarnesG(4*n)) * Gamma(n) * (1 - Gamma(3*n-2)*Gamma(3*n-1)/(Gamma(2*n-2)*Gamma(4*n-1))) / (BarnesG(3*n) * sqrt(2*Gamma(2*n))).
a(n) ~ exp(1/24) * 2^(8*n^2 - 5*n + 1/6) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 - 3*n + 5/12)), where A is the Glaisher-Kinkelin constant A074962. (End)

A367444 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the exchange principle, i.e., I(x, I(y,z)) = I(y, I(x,z)), for all x,y,z in L_n.

Original entry on oeis.org

1, 10, 165, 3863, 117096

Offset: 1

Marc Munar, Nov 18 2023

Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the exchange principle, 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)=n and I(n,0)=0 (discrete implication), and I(x, I(y,z)) = I(y, I(x,z)), for all x,y,z in L_n (exchange principle).

M. Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, Volume 621, 2023, 708-728.
M. Nachtegael and E. Kerre, Fuzzy logical operators on finite chains, International Journal of General Systems, Volume 29, 2000, 29-52.

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

A367445 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the contrapositive symmetry with respect to some discrete negation N, i.e., I(x,y) = I(N(y), N(x)), for all x,y in L_n.

Original entry on oeis.org

1, 8, 99, 2828, 152474

Offset: 1

Marc Munar, Nov 18 2023

Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the contrapositive symmetry with respect to some discrete negation 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)=n 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 with respect to a discrete negation N). A discrete negation N:L_n->L_n is a decreasing operator with N(0)=n and N(n)=0.

M. Munar, S. Massanet and D. Ruiz-Aguilera, A review on logical connectives defined on finite chains, Fuzzy Sets and Systems, Volume 462, 2023.

Particular case of the enumeration of discrete implications in general, enumerated in A360612.
The enumeration of discrete negations in general is given in A001700.
When the discrete negation is N(x)=n-x, for all x in L_n, the enumeration is given in A366540.

A367446 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the generalized modus ponens with respect to a discrete t-norm T, i.e., T(x,I(x,y))<=y, for all x,y in L_n.

Original entry on oeis.org

1, 9, 519, 150120, 202728377

Offset: 1

Marc Munar, Nov 18 2023

Number of discrete implications I:L_n^2->L_n defined on the finite chain L_n={0,1,...,n} satisfying the generalized modus ponens with respect to a discrete t-norm T, 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)=n and I(n,0)=0 (discrete implication), and T(x,I(x,y))<=y, for all x,y in L_n (generalized modus ponens with respect to a discrete t-norm T). A discrete t-norm T is a binary operator T:L_n^2->L_n such that T is increasing in each argument, commutative (T(x,y)=T(y,x) for all x,y in L_n), associative (T(x,T(y,z))=T(T(x,y),z) for all x,y,z in L_n) and has neutral element n (T(x,n)=x for all x in L_n).

Also, the number of discrete implications I satisfying the generalized modus tollens with respect to a discrete t-norm T and the classical discrete negation N_C, given by N_C(x)=n-x for all x in L_n, i.e., T(N(y),I(x,y)) <= N(x) for all x,y in L_n (generalized modus tollens with respect to a discrete t-norm T and a discrete negation N).

M. Munar, S. Massanet and D. Ruiz-Aguilera, A review on logical connectives defined on finite chains, Fuzzy Sets and Systems, Volume 462, 2023.

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

The enumeration of discrete negations in general is given in A001700.

A367447 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the law of importation with respect to a discrete t-norm T, i.e., I(T(x,y),z)=I(x,I(y,z)), for all x,y in L_n.

Original entry on oeis.org

1, 9, 102, 1529, 28702

Offset: 1

Marc Munar, Nov 18 2023

Number of discrete implications I:L_n^2->L_n defined on the finite chain L_n={0,1,...,n} satisfying the law of importation with respect to a discrete t-norm T, 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)=n and I(n,0)=0 (discrete implication), and I(T(x,y),z)=I(x,I(y,z)), for all x,y,z in L_n (law of importation with respect to a discrete t-norm T). A discrete t-norm T is a binary operator T:L_n^2->L_n such that T is increasing in each argument, commutative (T(x,y)=T(y,x) for all x,y in L_n), associative (T(x,T(y,z))=T(T(x,y),z) for all x,y,z in L_n) and has neutral element n (T(x,n)=x for all x in L_n).

M. Munar, S. Massanet and D. Ruiz-Aguilera, A review on logical connectives defined on finite chains, Fuzzy Sets and Systems, Volume 462, 2023.

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

A367540 Number of discrete implications I : L_n^2 -> L_n defined on the finite chain L_n = {0,1,...n} which satisfy the consequent boundary, i.e., I(x,y) >= y for all x,y in L_n.

Original entry on oeis.org

1, 8, 205, 17108, 4693632, 4253751084, 12768573248145, 127147160484338304, 4204352991963054866432

Offset: 1

Marc Munar, Nov 22 2023

Number of discrete implications I : L_n^2 -> L_n defined on the finite chain L_n={0,1,...n} satisfying the consequent boundary, 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) = n and I(n,0) = 0 (discrete implication), and I(x,y) >= y for all x,y in L_n (consequent boundary).

The proposed formula is recursive and implemented using dynamic programming using Python. Only the first 9 terms could be obtained. See GitHub link.

Marc Munar, Python program.
Marc Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, Volume 621, 2023, 708-728.
Marc Munar, S. Massanet and D. Ruiz-Aguilera, DiscreteFuzzyOperators - A Python library for computing with fuzzy operators, Zenodo, Version 1.13.

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

Python
```
See GitHub link
```

a(n) = Sum_{x in V_n'} G(v), where V_n' is the set of decreasing vectors v of n components whose entries are taken from L_n, v_1=n and v_i <= n-i+1 for all i in {2,...,n}, and G(v) is defined recursively as
   G(v) = det(A(v)) - Sum_{x in V_n(v)\v} G(v), where
       A(v)_{i,j} = binomial(n+v_j, n-i+j).
       V_n(v) is the set of decreasing vectors x of n components, whose entries are taken from L_n, and x_i <= v_i for all i in {1,...,n}.
   G(v) = binomial(n+k-1,k), if v=(k,0,...,0), with v being a vector of n components and 1 <= k <= n.

A367541 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the identity principle, i.e., I(x,x)=n for all x in L_n.

Original entry on oeis.org

1, 9, 246, 21307, 5967884, 5464753020, 16464650143150, 163867734760669875, 5401439489386802569500, 590665306641885854720733600, 214530897918187139967720562273920, 258998339526821950480574606267461843536, 1039917052871541256867935621512668719049634384, 13891789744852831118958512413787919060197070057215380

Offset: 1

Marc Munar, Nov 22 2023

Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n} satisfying the identity principle, 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)=n and I(n,0)=0 (discrete implication), and I(x,x)=n for all x in L_n (identity principle).

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

Particular case of the enumeration of discrete implications in general, enumerated in A360612.

Cf. A074962.

Table[(Product[Factorial[2 n - i + 1]/Factorial[n - 1 + i], {i, 1, n}]*Product[Product[(3 n + 3 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}] - Product[Factorial[2 n - i]/Factorial[n - 2 + i], {i, 1, n}]*Product[Product[(3 n + 1 - i + 2 j)/2, {j, 0, i - 2}], {i, 2, n}])*2^((n*(n - 1))/2)*Product[i^(n - i), {i, 1, n - 1}]*Product[1/Factorial[2 n + 1 - 2 i], {i, 1, n}], {n, 1, 15}]

a(n) = ((Product_{i=1..n} (2n-i+1)!/(n-1+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n+3-i+2j)/2)-(Product_{i=1..n} (2n-i)!/(n-2+i)!)*(Product_{i=2..n} Product_{j=0..i-2} (3n+1-i+2j)/2))*(2^(n*(n-1)/2))*(Product_{i=1..n} i^(n-i)/(2n+1-2i)!).

a(n) ~ exp(1/24) * 2^(2/3 + 5*n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 + 3*n + 5/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Nov 29 2023

cp's OEIS Frontend

A366540 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A367192 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the left neutrality principle, i.e., I(n,y)=y for all y in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A367434 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the ordering principle, i.e., I(x,y)=n if and only if x<=y, with x,y in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A367444 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the exchange principle, i.e., I(x, I(y,z)) = I(y, I(x,z)), for all x,y,z in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

A367445 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the contrapositive symmetry with respect to some discrete negation N, i.e., I(x,y) = I(N(y), N(x)), for all x,y in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

A367446 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the generalized modus ponens with respect to a discrete t-norm T, i.e., T(x,I(x,y))<=y, for all x,y in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

A367447 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the law of importation with respect to a discrete t-norm T, i.e., I(T(x,y),z)=I(x,I(y,z)), for all x,y in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

A367540 Number of discrete implications I : L_n^2 -> L_n defined on the finite chain L_n = {0,1,...n} which satisfy the consequent boundary, i.e., I(x,y) >= y for all x,y in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A367541 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the identity principle, i.e., I(x,x)=n for all x in L_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula