Some aspects of (r,k)-parking functions

An \emph{$(r,k)$-parking function} of length $n$ may be defined as a sequence $(a_1,\dots,a_n)$ of positive integers whose increasing rearrangement $b_1\leq\cdots\leq b_n$ satisfies $b_i\leq k+(i-1)r$. The case $r=k=1$ corresponds to ordinary parking functions. We develop numerous properties of $(r,k)$-parking functions. In particular, if $F_n^{(r,k)}$ denotes the Frobenius characteristic of the action of the symmetric group $\mathfrak{S}_n$ on the set of all $(r,k)$-parking functions of length $n$, then we find a combinatorial interpretation of the coefficients of the power series $\left( \sum_{n\geq 0}F_n^{(r,1)}t^n\right)^k$ for any $k\in \mathbb{Z}$. When $k>0$, this power series is just $\sum_{n\geq 0} F_n^{(r,k)} t^n$; when $k<0$, we obtain a dual to $(r,k)$-parking functions. We also give a $q$-analogue of this result. For fixed $r$, we can use the symmetric functions $F_n^{(r,1)}$ to define a multiplicative basis for the ring $\Lambda$ of symmetric functions. We investigate some of the properties of this basis

A useful variant of the Davis--Kahan theorem for statisticians

The Davis--Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. It relies on an eigenvalue separation condition between certain relevant population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, making it more natural and convenient for direct application in statistical contexts, and improving the bounds in some cases. We also provide an extension to situations where the matrices under study may be asymmetric or even non-square, and where interest is in the distance between subspaces spanned by corresponding singular vectors.Comment: 12 page

The lightest neutral hypernuclei with strangeness -1 and -2

Our current knowledge of the baryon--baryon interaction suggests that the dineutron $(n,n)$ and its strange analogue $(n,\Lambda)$ are unstable. In contrast, the situation is more favorable for the strange three-body system $(n,n,\Lambda)$, and even better for the four-body system $T\equiv (n,n,\Lambda,\Lambda)$ with strangeness $-2$, which is likely to be stable under spontaneous dissociation. The recent models of the hyperon-nucleon and hyperon-hyperon interactions suggest that the stability of the $(n,n,\Lambda)$ and $T$ is possible within the uncertainties of our knowledge of the baryon-baryon interactions. This new nucleus $T$ could be produced and identified in central deuteron--deuteron collisions via reaction $d+d\to T+K^++K^+$, and the tetrabaryon $T$ could play an important role in catalyzing the formation of a strange core in neutron stars.Comment: Revised version accepted by PR