Riordan paths and derangements
نویسندگان
چکیده
Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and (321, 31̄42)-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schröder numbers.
منابع مشابه
Cycles in the Graph of Overlapping Permutations Avoiding Barred Patterns
As a variation of De Bruijn graphs on strings of symbols, the graph of overlapping permutations has a directed edge π(1)π(2) . . . π(n+1) from the standardization of π(1)π(2) . . . π(n) to the standardization of π(2)π(3) . . . π(n + 1). In this paper, we consider the enumeration of d-cycles in the subgraph of overlapping (231, 41̄32)avoiding permutations. To this end, we introduce the notions of...
متن کاملGeneralized Narayana Polynomials, Riordan Arrays, and Lattice Paths
We study a family of polynomials in two variables, identifying them as the moments of a two-parameter family of orthogonal polynomials. The coefficient array of these orthogonal polynomials is shown to be an ordinary Riordan array. We express the generating function of the sequence of polynomials under study as a continued fraction, and determine the corresponding Hankel transform. An alternati...
متن کاملA Generalization of the $k$-Bonacci Sequence from Riordan Arrays
In this article, we introduce a family of weighted lattice paths, whose step set is {H = (1, 0), V = (0, 1), D1 = (1, 1), . . . , Dm−1 = (1,m − 1)}. Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the k-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and ...
متن کاملRiordan group approaches in matrix factorizations
In this paper, we consider an arbitrary binary polynomial sequence {A_n} and then give a lower triangular matrix representation of this sequence. As main result, we obtain a factorization of the innite generalized Pascal matrix in terms of this new matrix, using a Riordan group approach. Further some interesting results and applications are derived.
متن کاملSemiorders and Riordan Numbers
In this paper, we define a class of semiorders (or unit interval orders) that arose in the context of polyhedral combinatorics. In the first section of the paper, we will present a pure counting argument equating the number of these interesting (connected and irredundant) semiorders on n + 1 elements with the nth Riordan number. In the second section, we will make explicit the relationship betw...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 308 شماره
صفحات -
تاریخ انتشار 2008