A combinatorial proof of andrews' smallest parts partition function

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• Publié par: opakito
• Nombre de lectures: 151
• Langue: Français

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A Combinatorial Proof of Andrews’ Smallest Parts Partition Function

Kathy Qing Ji Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071, P.R. China [e-mail]

Submitted: Jan14, 2008; Accepted:Mar 19, 2008; Published: Apr 10, 2008 Mathematics Subject Classification: 05A17, 11P81

Abstract We give a combinatorial proof of Andrews’ smallest parts partition function with the aid of rooted partitions introduced by Chen and the author.

1 Introduction We adopt the common notation on partitions as used in [1].A partitionλof a positive integernis a finite nonincreasing sequence of positive integers λ= (λ 1 , λ 2 , . . . ,λ r ) P r such thatλ i =n.Thenλ i are called the parts ofλnumber of parts of. Theλis i=1 called the length ofλ, denoted byl(λ).The weight ofλis the sum of parts, denoted by |λ|.We letP(n) denote the set of partitions ofn. Letspt(n) denote the number of smallest parts in all partitions ofnandn s (λ) denote the number of the smallest parts inλ, we then have X spt(n) =n s (λ).(1.1) λ∈P(n) Below is a list of the partitions of 4 with their corresponding number of smallest parts. We see thatspt(4) = 10. λ∈ P(4)n s (λ) (4) 1 (3,1) 1 (2,2) 2 (2,1,1) 2 (1,1,1,1) 4

the electronic journal of combinatorics15(2008), #N12

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