Combinatorial Methods In The Theory Of Partitions

The theory of partition was studied from the eighteenth century by Euler, and developedby Cayley, Gauss, Hardy, Jacobi, Lagrange, Legendre, Littlewood, Rademacher, Ramanu-jan, Schur, Sylvester and MacMahon. Nowadays the theory of partition still attracts manymathematicians. So far lots of new partition theorems were found and proved (with newmethods) by many mathematicians. We can not list their names one by one. Among thesemathematicians Andrews as a leader of modern partition theorem have a great contributionto make this filed more substantial.Almost all partition theorems have relation with combinatorial identities or basic hy-pergeometric series. One of the most famous identities is the Rogers-Ramanujan identities(1.1.2) and (1.1.3), which can be stated in partition theory [60, Ch. 3].The discovery of the partition-theoretic statement of the Rogers-Ramanujan identitiesby MacMahon and Schur [69] prompted a search for similar partition theorems with a dif-ference other than 2 required of the summands in one class of the partitions considered.In 1926, Shcur [70] proved Theorem 3.1.1. Gleissberg [45] extended Schur's theorem togeneral modulus m > 3 (Theorem 3.1.2) in 1928. Go¨llnitz [46] proved his theorem of parti-tion in 1967, which can be seemed as a three residue classes extension of Schur's theorem.Alladi, Andrews, and Berkovich [3] proved a four parameter key identity, of which a deeperpartition theorem (Theorem 6.2.1) is a consequence. Theorem 6.2.1 can be seemed as anext level extend of theorem of Go¨llnitz.We first show that the formalism of overpartitions gives a simple involution [34] forthe product definition of the Gaussian coefficients. In the formulation of our involution, anoverline in the representation of an overpartition is endowed with a weight.We alos give a simply involution [35] on a key identity of Gleissberg theorem, which isa generalization of Schur's theorem. The Joichi-Stanton's Insertion Algorithm and overpar-tition were also employed.Then we concentrate on construction of bijection on two certain kind sets of partitionsof Shcur type [35]. First we give a direct bijection on two sets involved in the theoremof Go¨llnitz, and extend the bijection to a generalized form given by Alladi, Andrews, andGordon [4]. Then we give a deeper generalization of Go¨llnitz's theorem.Some related problems were list in section 6. The theorems list in section 6 have beenproved. We seek for direct bijective proofs. In the end of this section, we partially gener-alized one of Andrews's partitions theorem [13] and obtain a new one with respect to partsmodular kr where k ≥2, r ≥2 by MacMahon Modular diagram [61].In the last section we show a conjecture on unimodality and log-concavity of a kindq-Euler polynomials An(q), which is a q-analogue of up-down permutations.