On the difference partition and the stairway partition

Abstract

In this paper, for two arbitrary nonnegative sequences $\alpha=(a_1, a_2, a_3, …)$ and $\beta=(b_1, b_2, b_3,…)$ with $a_i \leg b_i$ (i=1, 2, 3, …), we define the sets of (m, n)-stairway partition $S_\alpha^\beta$, (m, n)-difference partition $D_\alpha^\beta$ and its restricted partitions. Specially, we are interested in $D_m$, $S_m$, $D^n$ and $S^n$. We give a combinatorial bijection φ to prove that the partitions of n in $D_\alpha^\beta$ are equinumerous with the partitions of n in $S_\alpha^\beta$. In addition, we can know that the alternating sum $|λ|_a$ of λ is equal to o(φ(λ)) in φ(λ) by the definition of $\phi$ where $|\lambda|_a$ stands for the alternating sum of λ, i.e. $|\lambda|_a=\sum_{i=1}^n(-1)^{i+1}\lambda_i$ and o(λ) the number of odd parts in λ. Thus we can get the following result \begin{displaymath} \sum_{\lambda\in D_\palpha^\beta} q^{|\lambda|{x^}|\lambda|_a} = \sum_{\lambda\in s_\alpha^\beta} q^{|\lambda|{x^}{o\lambda)} \end{displaymath} Afterward, using techniques in partition theory [1], we get the generating functions for $D_m$ and $S_m$. Then we obtain a new generalized identity related to Rogers-Ramanujan identity. We get the generating functions for $S^,n$, $S_m^n$ to obtain the generating functions for $D^n$, $D_m^n$. In addition, we get the generating functions for $D_\alpha^\beta$ and $S_\alpha^\beta$ which generalize above generating functions. After defining $\widetilde D_\alpha^\beta$ and $\widetilde S_\alpha^\beta$, we do the same jobs for $\widetilde D_\alpha^\beta$ and $\widetilde S_\alpha^\beta$ to obtain a new generalized identity related to another Rogers-Ramanujan identity. Finally, we discuss the Rogers-Ramanujan identities with above results.