اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدm-Level rook placements
61
0
0.0
(
0
)
تأليف
Kenneth Barrese
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of $m$ rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We generalize this result to any Ferrers board as well as giving a p,q-analogue. We also consider a dual situation involving weighted file placements which permit more than one rook in the same row. In both settings, we discuss properties of the resulting equivalence classes such as the number of elements in a class. In addition, we prove analogues of a theorem of Foata and Schutzenberger giving a distinguished representative in each class as well as make connections with the q,t-Catalan numbers. We end with some open questions raised by this work.
قيم البحث
اقرأ أيضاً
In this paper we answer a question posed by R. Stanley in his collection of Bijection Proof Problems (Problem 240). We present a bijective proof for the enumeration of walks of length $k$ a chess rook can move along on an $mtimes n$ board starting and ending on the same square.
The rook monoid $R_n$ is the finite monoid whose elements are the 0-1 matrices with at most one nonzero entry in each row and column. The group of invertible elements of $R_n$ is isomorphic to the symmetric group $S_n$. The natural extension to $R_n$
of the Bruhat-Chevalley ordering on the symmetric group is defined in cite{Renner86}. In this paper, we find an efficient, combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also give a useful, combinatorial formula for the length function on $R_n$.
The emph{simplicial rook graph} SR(d,n) is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency
and Laplacian matrices of SR(3,n) have integral spectrum for every $n$. The proof proceeds by calculating an explicit eigenbasis. We conjecture that SR(d,n) is integral for all $d$ and $n$, and present evidence in support of this conjecture. For $n<binom{d}{2}$, the evidence indicates that the smallest eigenvalue of the adjacency matrix is $-n$, and that the corresponding eigenspace has dimension given by the Mahonian numbers, which enumerate permutations by number of
Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index, $m$. Such polygons are called emph{$m$-convex} polygons and are characterised by having up to $
m$ indentations in the side. We use a `divide and conquer approach, factorising 2-convex polygons by extending a line along the base of its indents. We then use the inclusion-exclusion principle, the Hadamard product and extensions to known methods to derive the generating functions for each case.
A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some general conditi
ons on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of R-perfect poset codes for some R in the case of the crown poset and in the case of the union of disjoin chains. Index terms: perfect codes, poset codes
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
