tailieunhanh - Báo cáo toán học: "Ascent sequences and upper triangular matrices containing non-negative integers"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Ascent sequences and upper triangular matrices containing non-negative integers. | Ascent sequences and upper triangular matrices containing non-negative integers Mark Dukes Robert Parviainen Mathematics Division Science Institute University of Iceland 107 Reykjavik Iceland dukes@ The Mathematics Institute School of Computer Science Reykjavik University 103 Reykjavik Iceland parviainen@ Submitted Jan 25 2010 Accepted Mar 22 2010 Published Mar 29 2010 Mathematics Subject Classifications 05A05 05A19 Abstract This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics on these structures under this bijection and prove that some of these statistics are equidistributed. Several special classes of matrices are shown to have simple formulations in terms of ascent sequences. Binary matrices are shown to correspond to ascent sequences with no two adjacent entries the same. Bidiagonal matrices are shown to be related to order-consecutive set partitions and a simple condition on the ascent sequences generate this class. 1 Introduction Let Intn be the collection of upper triangular matrices with non-negative integer entries which sum to n G N such that all rows and columns contain at least one non-zero entry. For example Int3 Both authors were supported by grant no. 090038011 from the Icelandic Research Fund. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R53 1 We use the standard notation a b for the interval of integers a a 1 . b and define n 1 n . Given a sequence of integers y y1 . yn we say that y has an ascent at position i if yi yi 1. The number of ascents of y is denoted by asc y . Let An be the collection of ascent sequences of length n An x1 . xn xi G 0 1 asc x1 . xi-1 for all 1 i n where x1 0 and asc x1 0. For example A3 0 0 0 0 0 1 0 1 0 0 1 1 0 1 2 . These sequences were introduced in the recent paper by Bousquet-Melou et al. 1 and were shown to unify three

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