On the superimposition of Christoffel words

Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be reformulated as follows: for a $k$-letter alphabet A, with a fixed $k \geq 3$, there exists a unique balanced infinite word, up to letter permutations and shifts, that has mutually distinct letter frequencies. Motivated by the Fraenkel conjecture, we study in this paper whether two Christoffel words can be superimposed. Following from previous works on this conjecture using Beatty sequences, we give a necessary and sufficient condition for the superimposition of two Christoffel words having same length, and more generally, of two arbitrary Christoffel words. Moreover, for any two superimposable Christoffel words, we give the number of different possible superimpositions and we prove that there exists a superimposition that works for any two superimposable Christoffel words. Finally, some new properties of Christoffel words are obtained as well as a geometric proof of a classic result concerning the money problem, using Christoffel words

P-partitions and a multi-parameter Klyachko idempotent

Because they play a role in our understanding of the symmetric group algebra, Lie idempotents have received considerable attention. The Klyachko idempotent has attracted interest from combinatorialists, partly because its definition involves the major index of permutations. For the symmetric group S_n, we look at the symmetric group algebra with coefficients from the field of rational functions in n variables q_1,..., q_n. In this setting, we can define an n-parameter generalization of the Klyachko idempotent, and we show it is a Lie idempotent in the appropriate sense. Somewhat surprisingly, our proof that it is a Lie element emerges from Stanley's theory of P-partitions.Comment: 16 pages, 1 figure. Final version: incorporates suggestions of the referee, no changes to the result

Number of right ideals and a qq-analogue of indecomposable permutations

We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field $\mathbb F\_q$ is equal to $(q-1)^{n+1} q^{\frac{(n+1)(n-2)}{2}}\sum\_\theta q^{inv(\theta)}$, where the sum is over all indecomposable permutations in $S\_{n+1}$ and where $inv(\theta)$stands for the number of inversions of $\theta$.Comment: submitte

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions indexed by compositions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials.Comment: 30 page

Partition complexes, duality and integral tree representations

We show that the poset of non-trivial partitions of 1,2,...,n has a fundamental homology class with coefficients in a Lie superalgebra. Homological duality then rapidly yields a range of known results concerning the integral representations of the symmetric groups S_n and S_{n+1} on the homology and cohomology of this partially-ordered set.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-41.abs.htm

Liquidity costs: a new numerical methodology and an empirical study

We consider rate swaps which pay a fixed rate against a floating rate in presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit strategy optimizing an expected function of the hedging error. We here propose an efficient algorithm based on the stochastic gradient method to compute an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numerical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost