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

1,2

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

Table of n, a(n) for n=1..10.

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.

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.

Prog

(Python) See Github link

Crossrefs

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

Sequence in context: A156720 A288163 A268880 * A048143 A216420 A137083

Adjacent sequences: A367189 A367190 A367191 * A367193 A367194 A367195

Keyword

nonn,hard,more

Author

Marc Munar, Nov 09 2023

Status

approved