TAILIEUCHUNG - Báo cáo toán học: "An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding. | An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations Jason Bandlow and Kendra Killpatrick Mathematics Department Colorado State University Fort Collins Colorado bandlow@ killpatr@ Submitted July 23 2001 Accepted December 10 2001. MR Subject Classifications 05A15 05A19 Abstract The symmetric q t-Catalan polynomial Cn q t which specializes to the Catalan polynomial Cn q when t 1 was defined by Garsia and Haiman in 1994. In 2000 Garsia and Haglund described statistics a ft and b ft on Dyck paths such that Cn q t P qa tb where the sum is over all n X n Dyck paths. Specializing t 1 gives the Catalan polynomial Cn q defined by Carlitz and Riordan and further studied by Carlitz. Specializing both t 1 and q 1 gives the usual Catalan number Cn. The Catalan number Cn is known to count the number of n X n Dyck paths and the number of 312-avoiding permutations in Sn as well as at least 64 other combinatorial objects. In this paper we define a bijection between Dyck paths and 312-avoiding permutations which takes the area statistic a on Dyck paths to the inversion statistic on 312-avoiding permutations. The inversion statistic can be thought of as the number of 21 patterns in a permutation Ơ. We give a characterization for the number of 321 4321 . k 21 patterns that occur in Ơ in terms of the corresponding Dyck path. 1 Introduction The polynomial Cn q t called the q t-Catalan polynomial was introduced in 1994 by Garsia and Haiman 5 . They conjectured that it is the Hilbert series of the diagonal harmonic alternates and showed that it is the coefficient of the elementary symmetric function en in the symmetric polynomial DHn x q t the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. The polynomial is referred to as the q t-Catalan polynomial because specializing t 1 gives the q-Catalan polynomial THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R40 1 first given by Carlitz and Riordan 3 and

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