Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7, and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees 5 and 11 were combinatorially proved by Sandon and Zanello, and it remains open to find a combinatorial proof of the partition identity of degree 23. In this paper, we prove general colored partition identities with a restriction on the number of parts, which are connected to the partition identities arising from those modular equations. We also provide bijective proofs of these partition identities. In particular, one of these proofs gives bijective proofs of the partition identity of degree 23 for some cases, which also work for the identities of degrees 5 and 11 for the same cases.

Ramanujan的度数为3、5、7、11和23的模块化方程式产生漂亮的彩色分区标识。Warnaar对3和7级的模方程进行了解析推广，此后，作者找到了这些分区恒等式的双射证明，并于近期建立了对5、11和23度的模方程的解析归纳。 11由Sandon和Zanello组合证明，并且仍然可以找到度23的分区标识的组合证明。在本文中，我们证明了有色部件的一般性，但部件数量受到限制，这些部件与由这些模块化方程式产生的分区恒等式。我们还提供了这些分区身份的双射证明。尤其是，