Counting interesting elections

We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman--Stanley polytope of an n-tuple.Comment: 7 pages, 1 figure; published versio

Sub-committee Approval Voting and Generalised Justified Representation Axioms

Social choice is replete with various settings including single-winner voting, multi-winner voting, probabilistic voting, multiple referenda, and public decision making. We study a general model of social choice called Sub-Committee Voting (SCV) that simultaneously generalizes these settings. We then focus on sub-committee voting with approvals and propose extensions of the justified representation axioms that have been considered for proportional representation in approval-based committee voting. We study the properties and relations of these axioms. For each of the axioms, we analyse whether a representative committee exists and also examine the complexity of computing and verifying such a committee

Towards quantum-based privacy and voting

The privacy of communicating participants is often of paramount importance, but in some situations it is an essential condition. A typical example is a fair (secret) voting. We analyze in detail communication privacy based on quantum resources, and we propose new quantum protocols. Possible generalizations that would lead to voting schemes are discussed.Comment: 5 pages, improved description of the protoco

Random words, quantum statistics, central limits, random matrices

Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved [math.CO/9906120] that the expected shape \lambda of the semi-standard tableau produced by a random word in k letters is asymptotically the spectrum of a random traceless k by k GUE matrix. In this article we give two arguments for this fact. In the first argument, we realize the random matrix itself as a quantum random variable on the space of random words, if this space is viewed as a quantum state space. In the second argument, we show that the distribution of \lambda is asymptotically given by the usual local limit theorem, but the resulting Gaussian is disguised by an extra polynomial weight and by reflecting walls. Both arguments more generally apply to an arbitrary finite-dimensional representation V of an arbitrary simple Lie algebra g. In the original question, V is the defining representation of g = su(k).Comment: 11 pages. Minor changes suggested by the refere

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction