arXiv subjects

Jeffrey B. Remmel

Publications and source records attributed to Jeffrey B. Remmel.

22 records · Page 2Linked to original sources

arXiv2007

Counting descents, rises, and levels, with prescribed first element, in words

Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and generalize some of the results in [Counting occurrences of some subword patterns, Discrete Mathematics and Theoretical Computer Science 6 (2003), 001-012.].

math.CO↗

arXiv2007

The Combinatorics of the Garsia-Haiman Modules for Hook Shapes

Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.

math.RT↗

arXiv2006

Counting Descent Pairs with Prescribed Tops and Bottoms

Given sets X and Y of positive integers and a permutation sigma = sigma_1, sigma_2, ..., sigma_n in S_n, an X,Y-descent of sigma is a descent pair sigma_i > sigma_{i+1} whose "top" sigma_i is in X and whose "bottom" sigma_{i+1} is in Y. We give two formulas for the number P_{n,s}^{X,Y} of sigma in S_n with s X,Y-descents. P_{n,s}^{X,Y} is also shown to be a hit number of a certain Ferrers board. This work generalizes results of Kitaev and Remmel on counting descent pairs whose top (or bottom) is equal to 0 mod k.

math.CO↗

arXiv2003

On the Expressibility of Stable Logic Programming

(We apologize for pidgin LaTeX) Schlipf \cite{sch91} proved that Stable Logic Programming (SLP) solves all $\mathit{NP}$ decision problems. We extend Schlipf's result to prove that SLP solves all search problems in the class $\mathit{NP}$. Moreover, we do this in a uniform way as defined in \cite{mt99}. Specifically, we show that there is a single $\mathrm{DATALOG}^{\neg}$ program $P_{\mathit{Trg}}$ such that given any Turing machine $M$, any polynomial $p$ with non-negative integer coefficients and any input $σ$ of size $n$ over a fixed alphabet $Σ$, there is an extensional database $\mathit{edb}_{M,p,σ}$ such that there is a one-to-one correspondence between the stable models of $\mathit{edb}_{M,p,σ} \cup P_{\mathit{Trg}}$ and the accepting computations of the machine $M$ that reach the final state in at most $p(n)$ steps. Moreover, $\mathit{edb}_{M,p,σ}$ can be computed in polynomial time from $p$, $σ$ and the description of $M$ and the decoding of such accepting computations from its corresponding stable model of $\mathit{edb}_{M,p,σ} \cup P_{\mathit{Trg}}$ can be computed in linear time. A similar statement holds for Default Logic with respect to $Σ_2^\mathrm{P}$-search problems\footnote{The proof of this result involves additional technical complications and will be a subject of another publication.}.

cs.AI↗