We provide sufficient conditions under which the Catalan-like numbers are Stieltjes moment sequences. As applications, we show that many well-known counting coefficients, including the Bell numbers, the Catalan numbers, the central binomial coefficients, the central Delannoy numbers, the factorial numbers, the large and little Schr\"oder numbers, are Stieltjes moment sequences in a unified approach.Comment: 7 page

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Let G be a simple graph on the vertex set {v 1 , . . . , v n } with edge set E. Let K be a field. The graphical arrangement A G in K n is the arrangement (A)) of the cone c(A) contains a maximal chain of modular elements. The second author has shown that a graphical arrangement A G is supersolvable if and only if G is a chordal graph. He later considered a generalization of graphical arrangements which are called ψ-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a ψ-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free ψ-graphical arrangements.

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Let A be a hyperplane arrangement in A isomorphic to R n. Let V q be the q-Varchenko matrix for the arrangement A with all hyperplane parameters equal to q. In this paper, we consider three interesting cases of q-Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over Z[q].