A367192 - OEIS

%I #14 Nov 18 2023 08:37:49

%S 1,5,84,4719,884884,553361016,1153471856900,8012241391384695,

%T 185424118272842096128,461964068878932837522210816

%N 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.

%C 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).

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

%H Marc Munar, <a href="https://github.com/mmunar97/discrete-fuzzy-operators/blob/master/discrete_fuzzy_operators/counters/conjunctions/discrete_neutrality_principle_conjunctions_counter.py">Python program</a>.

%H Marc Munar, S. Massanet and D. Ruiz-Aguilera, <a href="https://doi.org/10.1016/j.ins.2022.10.121">On the cardinality of some families of discrete connectives</a>, Information Sciences, Volume 621, 2023, 708-728.

%H Marc Munar, S. Massanet and D. Ruiz-Aguilera, <a href="https://zenodo.org/doi/10.5281/zenodo.5031268">DiscreteFuzzyOperators - A Python library for computing with fuzzy operators</a>, Zenodo, Version 1.13.

%F a(n)=G((1,2,...,n)), where G(v) is defined recursively as:

%F    ·G(v)=det(A(v))-Sum_{x in V_n(v)\v} G(v), where:

%F        · A(v)_{i,j}=binomial(n+v_j, n-i+j).

%F        · 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}.

%F    ·G(v)=Binomial(n+x-1,x), if v=(x,0,...,0), with v being a vector of n components and 1<=x<=n.

%o (Python) See Github link

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

%K nonn,hard,more

%O 1,2

%A _Marc Munar_, Nov 09 2023