Dennis Eichhorn

Affiliation: University of California, Irvine

Email: deichhor@math.uci.edu

Title Of Talk: A New Decomposition of Partitions and its Arithmetic Implications.

Abstract: In 1990, Garvan, Kim, and Stanton used the decomposition of a partition into its $\ell$-core and $\ell$-quotient to give explicit cycles of partitions which give direct concrete combinatorial proofs of the first few Ramanujan congruences for $p(n)$. Recently, Breuer, Kronholm, and the speaker discovered a new decomposition of a partition into its ``$\ell$-box remainder" and ``$\ell$-box quotient" which allows for the construction of new cycles which provide direct concrete combinatorial proofs of both congruences and the periodicity of $p(n,d)$, the number of partitions of $n$ into parts of size at most $d$, modulo $M$. In this talk, we discuss these decompositions, the cycles of partitions they generate, and the arithmetic implications.

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