Daniel Warren
In this paper, a result of Albert, Atkinson, Handley, Holton, and Stromquist [Electron. J. Combin. 9 (2002), #R5] which characterizes the optimal packing behavior of the pattern 1243 is generalized in two directions. The packing densities of layered patterns of type (1^a,a) and (1,1,b) are computed.
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In this article we generalize packing density problems from permutations to patterns with repeated letters and generalized patterns. We are able to find the packing density for some classes of patterns and several other short patterns.
We consider maximum packings of edge-disjoint $4$-cliques in the complete graph $K_n$. When $n \equiv 1$ or $4 \pmod{12}$, these are simply block designs. In other congruence classes, there are necessarily uncovered edges; we examine the possible `leave' graphs induced by those edges. We give particular emphasis to the case $n \equiv 0$ or $3 \pmod{12}$, when the leave is $2$-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this case. We extend their construction and use several additional direct and recursive constructions to realize a variety of $2$-regular leaves. For va…
Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this question has been tangentially considered previously, this is the first systematic study of it. We find the packing density for various special sets of permutations and study permutation and pattern co-occurrence.
A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of quadruples $d(3,4,n)$ denotes the number of blocks in a maximum $3$-$(n,4,1)$ packing design, which is also the maximum number $A(n,4,4)$ of codewords in a code of length $n$, constant weight $4$, and minimum Hamming distance 4. In this paper the undecided 21 packing numbers $A(n,4,4)$ are shown to be equal to Johnson bound $J(n,4,4)$ $( =\lfloor\frac{n}{4}\lfloor\f…
Colourings of Uniform Group Divisible Designs and Maximum Packings
本文研究了均匀群可分设计的弱着色和最大打包，提出了块均衡 2-着色的构造方法，并推广了相关界到 $c>2$ 的情况。
A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations of m letters exist for every m \ge k^2/2+2k+1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain "permuted" packings of diamonds.
The integer $β(ρ, v, k)$ is defined to be the maximum number of blocks in any $(v, k)$-packing in which the maximum partial parallel class (or PPC) has size $ρ$. This problem was introduced and studied by Stinson for the case $k=3$. Here, we mainly consider the case $k = 4$ and we obtain some upper bounds and lower bounds on $β(ρ, v, 4)$. We also provide some explicit constructions of $(v,4)$-packings having a maximum PPC of a given size $ρ$. For small values of $ρ$, the number of blocks of the constructed packings are very close to the upper bounds on $β(ρ, v, 4)$. Some of our methods are ext…
We bound several quantities related to the packing density of the patterns 1(L+1)L...2. These bounds sharpen results of Bóna, Sagan, and Vatter and give a new proof of the packing density of these patterns, originally computed by Stromquist in the case L=2 and by Price for larger L. We end with comments and conjectures.
Packing Designs with large block size
本文研究了具有大块大小的 packing design，证明了在特定不等式条件下 packing number 的精确值，并基于块排序构造了 directed packing design 以产生 $(k-t)$-插入/删除码。
In this paper we investigate combinatorial constructions for $w$-cyclic holely group divisible packings with block size three (briefly by $3$-HGDPs). For any positive integers $u,v,w$ with $u\equiv0,1~(\bmod~3)$, the exact number of base blocks of a maximum $w$-cyclic $3$-HGDP of type $(u,w^v)$ is determined. This result is used to determine the exact number of codewords in a maximum three-dimensional $(u\times v\times w,3,1)$ optical orthogonal code with at most one optical pulse per spatial plane and per wavelength plane.
We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2),... n(k),.... Extending and improving previous work of Nurmela and Ostergard where previous patterns for n = n(k) of the form k*k, k*k-1, k*k-3, k(k+1), and 4k*k+k were observed, we identify new patterns for n = k*k-2 and n = k*k+[k/2]. We also find denser packings than those in [Nurmela, Ostergard] for n =21, 28, 34, 40, 43, 44, 45, and 47. In addition, we produce what we conjecture to be optimal packings for n =51, 52, …
The construction of optimal line packings in real or complex Euclidean spaces has shown to be a tantalizingly difficult task, because it includes the problem of finding maximal sets of equiangular lines. In the regime where equiangular lines are not possible, some optimal packings are known, for example, those achieving the orthoplex bound related to maximal sets of mutually unbiased bases. In this paper, we investigate the packing of subspaces instead of lines and determine the implications of maximality in this context. We leverage the existence of real or complex maximal mutually unbiased b…
Minimal abundant packings and choosability with separation
本文研究了 $(v,k,t)$ 填充的充裕性，改进了 Johnson 界并渐近确定了存在充裕填充的最小 $v$，据此证明了分离列表着色参数 $χ_\ell(K_n,c)/\sqrt{cn}$ 的极限存在且为 1，并对无穷多个 $n$ 给出了精确值。
We introduce and study a new notion of patterns in Stirling and $k$-Stirling permutations, which we call block patterns. We prove a general result which allows us to compute generating functions for the occurrences of various block patterns in terms of generating functions for the occurrences of patterns in permutations. This result yields a number of applications involving, among other things, Wilf equivalence of block patterns and a new interpretation of Bessel polynomials. We also show how to interpret our results for a certain class of labeled trees, which are in bijection with Stirling pe…
Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise to optimal frequency-hopping sequences and optimal comma-free codes. New constructions for PCDPs, based on almost difference sets and cyclic difference matrices, are given. These produce new infinite families of optimal PCDPs (and hence optimal frequency-hopping sequences and optimal comma-free codes). The existence problem for optimal PCDPs in ${\mathbb Z}_{3m}$, with $m$ base blocks of size three, is also solved for all $m\not\equiv 8,16\pmod{24}$.
A Faster Deterministic Algorithm for Mader's $\mathcal{S}$-Path Packing
针对Mader's $\mathcal{S}$-路径打包问题，提出一种新的确定性算法，通过归约到线性拟阵奇偶性，利用增广路径算法并优化矩阵运算，时间复杂度达$O(mnk)$，优于现有结果。
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized clust…
To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each block and blocks arranged in increasing order of their first entries--we count the partitions of [n] whose flattening avoids a single 3-letter pattern. Five counting sequences arise: a null sequence, the powers of 2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of the Catalan numbers.
On the maximum rank of Toeplitz block matrices of blocks of a given pattern
研究了块下三角Toeplitz块矩阵的最大秩问题，证明在块满足特定结构条件时，其最大秩等于术语秩。
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