On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.Comment: 19 page

Some remarks on sign-balanced and maj-balanced posets

Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the work of Frank Ruskey, who was interested in the efficient generation of all linear extensions of P. We survey a number of techniques for showing that posets are sign-balanced, and more generally, computing their "imbalance." There are close connections with domino tilings and, for certain posets, a "domino generalization" of Schur functions due to Carre and Leclerc. We also say that P is maj-balanced if exactly half the linear extensions of P have even major index. We discuss some similarities and some differences between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and Conjecture 3.6 has been adde

Entanglement Content of Quantum Particle Excitations II. Disconnected Regions and Logarithmic Negativity

In this paper we study the increment of the entanglement entropy and of the (replica) logarithmic negativity in a zero-density excited state of a free massive bosonic theory, compared to the ground state. This extends the work of two previous publications by the same authors. We consider the case of two disconnected regions and find that the change in the entanglement entropy depends only on the combined size of the regions and is independent of their connectivity. We subsequently generalize this result to any number of disconnected regions. For the replica negativity we find that its increment is a polynomial with integer coefficients depending only on the sizes of the two regions. The logarithmic negativity turns out to have a more complicated functional structure than its replica version, typically involving roots of polynomials on the sizes of the regions. We obtain our results by two methods already employed in previous work: from a qubit picture and by computing four-point functions of branch point twist fields in finite volume. We test our results against numerical simulations on a harmonic chain and find excellent agreement

Prime arithmetic Teichmuller discs in H(2)

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two all translation surfaces in H(2) tiled by a prime number n > 3 of squares fall into exactly two Teichmuller discs, only one of them with elliptic points, and that the genus of these discs has a cubic growth rate in n.Comment: Accepted for publication in Israel Journal of Mathematics. A previous version circulated with the title "Square-tiled surfaces in H(2)''. Changes from v1: improved redaction, fixed typos, added reference