On Partition Classes Arising from Parity, Differences, and Repeated Smallest Parts
Rahul Kumar, Nargish Punia
TL;DR
The paper studies partition classes defined by parity and repetition of the smallest part, extending Euler's partition theorem to new families. It employs a dual approach of bijective combinatorics and analytic $q$-series to connect $A(n)$, $B(n)$, $C(n)$, $D(n)$ with refinements like $D_k(n)$, $B_k^{e/o}(n)$, and $C_k^{e/o}(n)$, deriving explicit generating functions such as $\sum_{n\ge0} D_k(n) q^n=\sum_{n\ge0} q^{nk}(-q^{n+1};q)_\infty$ and key equalities $A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1)$ and $A_k(n)=\frac{1}{2}D_k(n+1)$. It provides bijective proofs for these correspondences (e.g., $A_k(n)=\frac{1}{2}D_k(n+1)$, $B_k^e(n)=C_k^e(n+1)$, $B_k^o(n)=C_k^o(n+1)$) in Sections 2, and complementary analytic proofs via $q$-series in Section 3, including a Legendre-type analogue and finite-analogue identities. The results broaden Euler’s theorem by linking parity-difference partitions to repeated-smallest-part partitions and offer generating-function tools and parity-structural insights that motivate future bijective work and extensions to finite analogues. Overall, the work deepens structural understanding in partition theory and sets a program for explicit bijections in remaining cases and for broader analogues.
Abstract
In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities connecting these various classes of partitions. Moreover, our identities help us to extend the Euler's partition theorem. An analogue of Legendre's theorem of the partition-theoretic interpretation of Euler's pentagonal number theorem is also derived. Both combinatorial and $q$-series proofs are given for our results.