Bijective Proofs of Certain Vector Partition Identities

نویسنده

  • BRUCE E. SAGAN
چکیده

for 1 <; i < k. L. Carlitz [2] first derived the generating function for restricted bipartite partitions. Subsequently Carlitz and Roselle [3] enumerated certain special families of these partitions e.g., restricted bipartite partitions where the m< and % are all odd. Finally both Roselle [4] and Andrews [1] have obtained different generalizations for multipartite partitions. All these results have been proved by manipulation of formal power series. However, the simplicity of the generating functions obtained suggests that purely combinatorial methods could be applied. The purpose of this paper is to give direct bijective proofs which, in addition, permit us to count a new family of partitions.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Permutation Statistics and q-Fibonacci Numbers

In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of q-Fibonacci numbers, which they related to q-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern r...

متن کامل

Bijective Proofs of Partition Identities of MacMahon, Andrews, and Subbarao

We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to 2, 3, 4, 6 (mod 6), together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result.

متن کامل

Finding direct partition bijections by two-directional rewriting techniques

One basic activity in combinatorics is to establish combinatorial identities by so-called ‘bijective proofs,’ which consists in constructing explicit bijections between two types of the combinatorial objects under consideration. We show how such bijective proofs can be established in a systematic way from the ‘lattice properties’ of partition ideals, and how the desired bijections are computed ...

متن کامل

Bijective Proofs of a Theorem of Fine and Related Partition Identities

In this paper, we prove a theorem of Fine bijectively. Stacks with summits and gradual stacks with summits are also discussed.

متن کامل

Partition Bijections, a Survey

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2004