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 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.
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.
FORMULA
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.
PROG
(Python) See GitHub link
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Marc Munar, Nov 22 2023
STATUS
approved