TAILIEUCHUNG - Báo cáo toán học: "Concave Compositions"

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: Concave Compositions. | Concave Compositions George E. Andrews Department of Mathematics The Pennsylvania State University University Park PA 16802 andrews@ Submitted Jul 20 2010 Accepted Apr 12 2011 Published Apr 23 2011 Mathematics Subject Classification 05A17 Dedicated to my friend Doron Zeilberger. Abstract Concave compositions are compositions . ordered partitions of a number in which the parts decrease up to the middle summand s and increase thereafter. Perhaps the most surprising result is for even length concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function discoveries also lead to new facts about concatenatable spiral self-avoiding walks. 1 Introduction Concave compositions are situated between ordinary integer partitions where 3 3 1 and 3 1 3 are considered the same partition of 7 and integer compositions where 3 3 1 and 3 1 3 are considered two different compositions of 7 . Concave compositions of even length CCELs 2m are sums of the form 22 ai V bi where ai Ũ2 am bm bm-1 bl where am 0. Concave compositions of odd length 2m 1 come in two flavors. First CCOL1s are sums of the form 22 ai 22 bi ai a2 am 1 bm bm-1 bl where am 1 0. Partially supported by National Science Foundation Grant DMS-0801184 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P6 1 Second CCOL2s a1 a2 am 1 bm bm-1 b1 where am 1 E 0. We denote by ce n co1 n and co2 n the number of CCELs CCOLls CCOL2s respectively that sum to n. The related generating functions are CE q Vn ce n qn CO1 q En 0 c 1 n qra and CO2 q En 0 co2 n qn. Theorem 1. For q 1 1 E qn 3n-1 2 1 - qn n 1 CE q 1 1 nqn 3n-1 2 1 qn n 1 1 q q1 2 q5 q7 q12 q15 1 q q2 q5 q7 q12 q15 1 2q2 2q3 4q4 4q5 8q6 . By the way of illustrations we note that the eight CCELs of 6 are 5001 1005 4002 2004 3003 33 2112 210012. Theorem 2. For q 1 1 q2n 3n-1 1 q4n CO1 q 1 1 nqn 3n-1 2 1 qn n 1 1 q4 q8 q20 q28 q48 q60 1 q q2 q5 q7 q12 .

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