Abstract
Fuss-Catalan number is a family of generalized Catalan numbers. We begin by two definitions of Fuss-Catalan numbers and some basic properties. And we give some combinatorial interpretations different from original Catalan numbers. Finally we generalize the Jonah's theorem as its applications.
Highlights
Catalan numbers {cn}n≥0 1 are said to be the sequence satisfying the recursive relation cn 1 c0cn c1cn−1 · · · cnc[0], c0 1.It is well known that the nth term of Catalan numbers is cn 1/ n 2n n 1/2n 1 n and {cn}n≥0{1, 1, 2, 5, 14, 42, 132, . . .}
One of many combinatorial interpretations of Catalan numbers is that cn is the number of shortest lattice paths from 0, 0 to n, n on the 2-dimensional plane such that those paths lie beneath the line y x
Fuss-Catalan numbers {cns }s,n≥0 were investigated by Fuss 2 and studied by several authors 1, 3–7
Summary
Catalan numbers {cn}n≥0 1 are said to be the sequence satisfying the recursive relation cn 1 c0cn c1cn−1 · · · cnc[0], c0 1. One of many combinatorial interpretations of Catalan numbers is that cn is the number of shortest lattice paths from 0, 0 to n, n on the 2-dimensional plane such that those paths lie beneath the line y x. 3 cns is the number of shortest lattice paths from 0, 0 to n, s − 1 n on the 2-dimensional plane such that those paths lie beneath y s − 1 x. The proposition describing Fuss-Catalan numbers could be restated in the language of generating functions. There are many combinatorial interpretations of Fuss-Catalan numbers, but most of them are similar to that of Catalan numbers. Hilton and Pedersen 9 generalized an identity called Jonah’s theorem which involves Catalan numbers.
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