1. [2] Find the number f (n) of pairs (λ, µ) such that λ ⊢ n and µ covers λ in Young's lattice Y. Express your answer in terms of p(k), the number of partitions of k, for certain values of k. Try to give a direct bijection, avoiding generating functions, recurrence relations, induction, etc.

The aim of this course is to tour the highlights of arithmetic combinatorics the combinatorial estimates relating to the sums, differences, and products of finite sets, or to related objects such as arithmetic progressions. The material here is of course mostly combinatorial, but we will also exploit the Fourier transform at times. We will also discuss the recent applications of this theory to ...

We express the genus-two fixed-complex-structure enumerative invariants of P and P in terms of the genus-zero enumerative invariants. The approach is to relate each genus-two fixedcomplex-structure enumerative invariant to the corresponding symplectic invariant.

Gian-Carlo Rota believed that mathematics is a unity, in the deep sense that the same themes recur – as analogies – in its many branches. Thus, it comes as no surprise that Rota perceived combinatorial themes in continuous mathematics (as well as the other way around). In the preface to his book Introduction to Geometric Probability [10], written with the first author, Rota suggested only partl...

Many applications often require finding sets of entities of interest that meet certain constraints. Such set-based queries (SQs) can be broadly classified into two types: optimization SQs that involve some optimization constraint and enumerative SQs that do not have any optimization constraint. While there has been much research on the evaluation of optimization SQs, there is very little work o...