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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.

%I A367540 #8 Nov 25 2023 08:42:00
%S A367540 1,8,205,17108,4693632,4253751084,12768573248145,127147160484338304,
%T A367540 4204352991963054866432
%N 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.
%C A367540 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).
%C A367540 The proposed formula is recursive and implemented using dynamic programming using Python. Only the first 9 terms could be obtained. See GitHub link.
%H A367540 Marc Munar, <a href="https://github.com/mmunar97/discrete-fuzzy-operators/blob/master/discrete_fuzzy_operators/counters/conjunctions/discrete_consequent_boundary_conjunctions_counter.py">Python program</a>.
%H A367540 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 A367540 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 A367540 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
%F A367540    G(v) = det(A(v)) - Sum_{x in V_n(v)\v} G(v), where
%F A367540        A(v)_{i,j} = binomial(n+v_j, n-i+j).
%F A367540        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 A367540    G(v) = binomial(n+k-1,k), if v=(k,0,...,0), with v being a vector of n components and 1 <= k <= n.
%o A367540 (Python) See GitHub link
%Y A367540 Particular case of the enumeration of discrete implications in general, enumerated in A360612.
%K A367540 nonn,hard,more
%O A367540 1,2
%A A367540 _Marc Munar_, Nov 22 2023