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.

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

Offset: 1

Author

Marc Munar, Nov 09 2023

Keywords

• nonn
• hard
• more

Comments

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.

Links

• 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.

Crossrefs

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

Programs

Python

See Github link

Formula

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.