Partition theorems and the Chinese remainder theorem

The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$ with all parts repeated at least once equals the number of partitions of $n$ where all parts must be even or congruent to $3 \pmod 6$. These partition theorems were further extended by Glaisher, Andrews, Subbarao, Nyirenda and Mugwangwavari. In this paper, we utilize the Chinese remainder theorem to prove a comprehensive partition theorem that encompasses all existing partition theorems. We also give a natural generalization of Euler's theorem based on a special complete residue system. Furthermore, we establish interesting congruence connections between the partition function $p(n)$ and related partition functions

Partitions of the Rogers-Ramanujan Type

U ovom radu bavimo se particijama Rogers-Ramanujanovog tipa. Particija broja n je rastav od n na sumande, gdje poredak nije važan. Particije grafički reprezentiramo pomoću Youngovog dijagrama i taj prikaz koristimo za dokazivanje identiteta za particijsku funkciju. Broj particija čiji su svi dijelovi neparni jednak je broju particija čiji su svi dijelovi medusobno različiti. Lijeva strana prvog i drugog Rogers-Ramanujanovog identiteta je funkcija izvodnica za 2-različite i 2-različite particije s uvjetom da je najmanji dio veći ili jednak od 2. Takve particije nazivamo particije Rogers-Ramanujanovog tipa. Omjer brojeva particija Rogers-Ramanujanovog tipa teži prema zlatnom rezu. Broj particija čija je apsolutna vrijednost razlike izmedu dva dijela veća ili jednaka od 2 jednak je broju particija kod kojih je najmanji parni dio, ukoliko postoji, dvostruko veći od broja neparnih dijelova te particije. Particije s tim svojstvima su poznate kao Rogers-Ramanujanove i polarizirane particije.In this paper we are dealing with partitions of the Rogers-Ramanujan’s type. The partition of n is an decomposition of n on sumande, where order is not important. Partitions are graphically displayed using Young diagram and we use this view to prove the identities for the partition function. The number of partitions whose parts are odd is equal to the number of partitions whose parts are distinct. The left side of the first and second RogersRamanujan’s identity is the 2-distinct and 2-distinct partition with the condition that the smallest part is greater or equal than 2. Such partitions are called the Rogers-Ramanujan type partitions. We observe the ratio of Rogers-Ramanujan’s partitions, where it is known that this ratio actually tends to the golden ratio. The number of 2-distinct partitions is equal to the number of 1-distant partitions whose the smallest even part, if there are any, are greater than twice the number of odd parts. Partitions with these properties are known as Rogers-Ramanujan’s and polarized partition

On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.Comment: 19 page