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Register a new userOn certain weighted 7-colored partitions
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Inspired by Andrews 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.
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In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis results on the rank and crank counts for ordinary partitions.
Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.
A generalized crank ($k$-crank) for $k$-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined $k$-crank. Namely, we first obtain some inequalities between the $k$-crank counts $M_{k}(r,m,n)$ for $m=2,3$ and $4$, then we prove the positivity of symmetrized even $k$-crank moments weighted by the parity for $k=2$ and $3$. We conclude with several remarks on furthering the study initiated here.
Motivated by Alladis recent multi-dimensional generalization of Sylvesters classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchys identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.
A colored space is the pair $(X,r)$ of a set $X$ and a function $r$ whose domain is $binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Zsubseteq X$. We shall write $Ysimeq_r Z$ if there exists a bijection $f:Yto Z$ such that $r(U)=r(f(U))$ for each $Uinbinom{Y}{2}$. We denote the numbers of equivalence classes with respect to $simeq_r$ contained in $binom{X}{2}$ and $binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)leq a_3(r)$ when $5leq |X|$, and show what happens when the equality holds.
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