Related papers: Longest increasing subsequences of random colored …
We introduce and study a new random permutation model that generalizes the $k$-card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce…
Let $LA_{n}(\tau)$ be the length of the longest alternating subsequence of a uniform random permutation $\tau\in[n]$. Classical probabilistic arguments are used to rederive the asymptotic mean, variance and limiting law of $LA_{n}(\tau)$.…
Let $\pi$ be a permutation of $[n]=\{1,\dots,n\}$ and denote by $\ell(\pi)$ the length of a longest increasing subsequence of $\pi$. Let $\ell_{n,k}$ be the number of permutations $\pi$ of $[n]$ with $\ell(\pi)=k$. Chen conjectured that the…
We study the length of $T$-contaminated runs of heads in the well-known coin tossing experiment. A $T$-contaminated run of heads is a sequence of consecutive heads interrupted by $T$ tails. For $T=1$ and $T=2$ we find the asymptotic…
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions…
Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$. We prove in this paper that there exists a universal constant $n_1$ such…
In this paper, we study two problems related to planar matchings in random bipartite graphs. First, we colour each edge of the complete bipartite graph $K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$ grows…
The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding…
Considering the optimal alignment of two i.i.d. random sequences of length $n$, we show that when the scoring function is chosen randomly, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges…
Edge-coloring problems with forbidden patterns are decision problems asking to find an edge-coloring of the input graph which avoids a homomorphism from a fixed forbidden family of edge-colored graphs. In the precolored version of these…
An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some…
We advance a probabilistic approach to the Hadwiger-Nelson problem initially developed by the Polymath16 project, in particular relating the approach to finite unit-distance graphs. We define the numerical \textit{badness} of a given…
Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…
In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most $b$ and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for…
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum…
In this work, we consider a variant of the classical Longest Common Subsequence problem called Doubly-Constrained Longest Common Subsequence (DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings, and a function Co…
In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while…
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…
We consider the cardinality of supercritical oriented bond percolation in two dimensions. We show that, whenever the origin is conditioned to percolate, the process appropriately normalized converges asymptotically in distribution to the…