A determinant property of Catalan numbers
نویسندگان
چکیده
Catalan numbers arise in a family of persymmetric arrays with determinant 1. The demonstration involves a counting result for disjoint path systems in acyclic directed graphs.
منابع مشابه
Some Identities and Formulas Involving Generalized Catalan Numbers Siu-ah Ng
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
متن کاملSome Aspects of Hankel Matrices in Coding Theory and Combinatorics
Hankel matrices consisting of Catalan numbers have been analyzed by various authors. DesainteCatherine and Viennot found their determinant to be ∏ 1≤i≤j≤k i+j+2n i+j and related them to the Bender Knuth conjecture. The similar determinant formula ∏ 1≤i≤j≤k i+j−1+2n i+j−1 can be shown to hold for Hankel matrices whose entries are successive middle binomial coefficients (2m+1 m ) . Generalizing t...
متن کاملSome New Binomial Sums Related to the Catalan Triangle
In this paper, we derive many new identities on the classical Catalan triangle C = (Cn,k)n>k>0, where Cn,k = k+1 n+1 ( 2n−k n ) are the well-known ballot numbers. The first three types are based on the determinant and the fourth is relied on the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combin...
متن کاملGeneralized Catalan numbers and the enumeration of planar embeddings
The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns. We give a new combinatorial interpretation for the Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the p-Catalan numbers in terms of p-ary t...
متن کاملPermanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences
In this paper we construct two types of Hessenberg matrices with the property that every weighted isobaric polynomial (WIP) appears as a determinant of one of them, and as the permanent of the other. Every integer sequence which is linearly recurrent is representable by (an evaluation of) some linearly recurrent sequence of WIPs. WIPs are symmetric polynomials written in the elementary symmetri...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Mathematics
دوره 211 شماره
صفحات -
تاریخ انتشار 2000