• Ghurumuruhan Ganesan

Articles written in Proceedings – Mathematical Sciences

• Multiplicity of Summands in the Random Partitions of an Integer

In this paper, we prove a conjecture of Yakubovich regarding limit shapes of `slices’ of two-dimensional (2D) integer partitions and compositions of 𝑛 when the number of summands $m\sim An^\alpha$ for some $A&gt;0$ and $\alpha &lt; \frac{1}{2}$. We prove that the probability that there is a summand of multiplicity 𝑗 in any randomly chosen partition or composition of an integer 𝑛 goes to zero asymptotically with 𝑛 provided 𝑗 is larger than a critical value. As a corollary, we strengthen a result due to Erdös and Lehner (Duke Math. J. 8(1941) 335–345) that concerns the relation between the number of integer partitions and compositions when $\alpha=\frac{1}{3}$.

• Traveling salesman problem across well-connected cities and with location-dependent edge lengths

Consider $n$ nodes $\{Xi\}_{1\leq i \leq n}$ distributed independently across $N$ cities located in the unit square $S$, each according to a certain distribution $g_{N}(\cdot)$. Each city is modelled as an $r_{n} \times r_{n}$ square and $\rm{TSPC}_{n}$ denotes the weight of the minimum weighted length cycle containing all the $n$ nodes, where the edge length between nodes $X_{i}$ and $X_{j}$ is location-dependent and based on a metric $d$ that is equivalent to the Euclidean metric. We obtain variance estimates for $\rm{TSPC}_{n}$ and prove that if the cities are well connected in a certain sense, then $\rm{TSPC}_{n}$ appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for $\rm{TSPC}_{n}$. Using the proof techniques, we also study results $\rm{TSP}_{n}$ of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square $S$ with location-dependent edge lengths. We obtain variance estimates and convergence in probability for $\rm{TSP}_{n}$ appropriately scaled and centred.

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