Generating function for K-restricted jagged partitions

Jagged partitions and lattice paths

A lattice-path description of $K$-restricted jagged partitions is presented. The corresponding lattice paths can have peaks only at even $x$ coordinate and the maximal value of the height cannot be larger than $K-1$. Its weight is twice that of the corresponding jagged partitions. The equivalence is demonstrated at the level of generating functions. A bijection is given between $K$-restricted jagged partitions and partitions restricted by the following frequencies conditions: $f_{2j-1}$ is even and $f_j+f_{j+1}\leq K-1$, where $f_j$ is the number of occurrences of the part $j$ in the partition…

Jagged partitions

By jagged partitions we refer to an ordered collection of non-negative integers $(n_1,n_2,..., n_m)$ with $n_m\geq p$ for some positive integer $p$, further subject to some weakly decreasing conditions that prevent them for being genuine partitions. The case analyzed in greater detail here corresponds to $p=1$ and the following conditions $n_i\geq n_{i+1}-1$ and $n_i\geq n_{i+2}$. A number of properties for the corresponding partition function are derived, including rather remarkable congruence relations. An interesting application of jagged partitions concerns the derivation of generating fun…

Enumeration of plane partitions with a restricted number of parts

In the paper, the quantum-statistical approach is used to estimate the number of restricted plane partitions of an integer $n$ with the number of parts not exceeding some finite $N$. The analogy between this number-theoretical problem and the enumeration of microstates of the ideal two-dimensional Bose-gas is used. The conjectured expression for the number of restricted plane partitions shows a good agreement between calculated and exact values for $n=10\div20$.

关于某些受限平面分拆和晶体熔化模型的一个注记

A remark on certain restricted plane partitions and crystal melting model

本文利用Okounkov、Reshetikhin和Vafa提出的晶体熔化模型，给出了两类平面分拆配分函数的计算公式，并推导出极限边界下平面分拆配分函数的乘积公式，证明了该配分函数与双P^1模型的开弦振幅等价，同时得到了z轴方向极限边界下对称平面分拆配分函数的乘积公式。

Gentile statistics and restricted partitions

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for the restricted or coloured partitions p_k^s(n), which is the number of partitions of an integer n into the summand of s^{th} powers of integers such that each power of a given integer may occur utmost k times. While the method is not rigorous, it reproduces the well known asymptotic results for s=1 apar…

The Euler-Riemann Gases, and Partition Identities

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice ($s$-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order $s$ of the parafermions. It is also shown that these pa…