Marc Munar - cp's OEIS frontend

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.

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

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.

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.

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.

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.

A367442 Number of discrete uninorms defined on the finite chain L_n={0,1,...,n-1,n}.

Original entry on oeis.org

2, 6, 22, 92, 426, 2146, 11634, 67472

Offset: 1

Marc Munar, Nov 18 2023

The number of discrete uninorms on the finite chain L_n={0,1,...,n-1,n}, i.e., the number of monotonic increasing binary functions U: L_n^2->L_n such that U is commutative (U(x,y)=U(y,x) for all x,y in L_n), associative (U(U(x,y),z)=U(x,U(y,z)) for all x,y,z in L_n) and has neutral element e in L_n (U(x,e)=x for all x in L_n).

M. Couceiro, J. Devillet and J.L. Marichal. Characterizations of idempotent discrete uninorms, Fuzzy Sets and Systems, Volume 334, 2018, 60-72.
M. Mas, S. Massanet, D. Ruiz-Aguilera, and J. Torrens A survey on the existing classes of uninorms, Journal of Intelligent and Fuzzy Systems, Volume 29, 2015, 1021-1037.
M. Munar, S. Massanet and D. Ruiz-Aguilera, On the cardinality of some families of discrete connectives, Information Sciences, Volume 621, 2023, 708-728.

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.

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)

A367432 Number of commutative discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n} that are smooth.

Original entry on oeis.org

1, 2, 10, 80, 1008, 19764, 600028, 28134464, 2034669118, 226781039624

Offset: 0

Marc Munar, Nov 18 2023

The number of smooth and commutative 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, F(x,y)=F(y,x) for all x,y in L_n (commutativity), and F(x+1,y)-F(x,y)<=1 and F(y,x+1)-F(y,x)<=1 for all y in L_n and x in L_n\{n} (smooth).

Also, the number of (n+1)X(n+1) integer symmetric matrices (m_{i,j}) such that m_{1,1}=1, m_{n+1,n+1}=n+1, and all rows and columns are (weakly) monotonic without jumps larger than 1.

Symmetric counterpart of matrices enumerated in A306372.

Smooth counterpart of operators defined in A366447.

a(0) and a(7)-a(9) from Martin Ehrenstein, Dec 01 2023

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.

cp's OEIS Frontend

User: Marc Munar

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.

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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.

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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.

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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.

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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.

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A367442 Number of discrete uninorms defined on the finite chain L_n={0,1,...,n-1,n}.

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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.

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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.

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A367432 Number of commutative discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n} that are smooth.

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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.

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