Combinatorics of counting finite fuzzy subsets

A study of a family of generating functions of Nelsen–Schmidt type and some identities on restricted barred preferential arrangements

“…for n, r ≥ 0 we have; The identity in the conjecture when r = 0 appears in [4] as equation 9. For r = 2 the identity in the conjecture occurs in [9] as an identity for the number of preferential fuzzy subsets of a set X n ; it occurs as equation 3.4 in [9].…”