arXiv subjects

Moriah Sigron

Publications and source records attributed to Moriah Sigron.

8 recordsLinked to original sources

arXiv2024

Interval Posets and Polygon Dissections

The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval posets of permutations. We present a bijection between tree interval posets and convex polygons with non-crossing diagonals, offering a novel geometric perspective on this purely combinatorial concept. Additionally, we provide an enumeration of interval posets using this bijection and demonstrate its application to block-wise simple permutations.

cs.DM↗

arXiv2023

Blockwise simple permutations

A permutation is called {\it {block-wise simple}} if it contains no interval of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of permutations and explore some of its combinatorial properties. We present a generating function for this set, as well as a recursive formula for counting block-wise simple permutations. Following Tenner, who founded the notion of interval posets, we characterize and count the interval posets corresponding to block-wise simple permutations. We also present a bijection between these interval posets and certain tiling's of the $n$-gon. Finally, we prove that the bi-variate distribution of the descent and inverse descent numbers are gamma-positive, provided the correctness of our recent conjecture on simple permutations.

math.CO↗

arXiv2020

On the poset of King-Non-Attacking permutations

A king-non-attacking permutation is a permutation $π\in S_n$ such that $|π(i)-π(i-1)|\neq 1$ for each $i \in \{2,\dots,n\}$. We investigate the structure of the poset of these permutations under the containment relation, and also provide some results on its Möbius function.

math.CO↗

arXiv2020

Counting King Permutations on the Cylinder

We call a permutation $σ=[σ_1,\dots,σ_n] \in S_n$ a {\em cylindrical king permutation} if $ |σ_i-σ_{i+1}|>1$ for each $1\leq i \leq n-1$ and $|σ_1-σ_n|>1$. We present some results regarding the distribution of the cylindrical king permutations, including some interesting recursions. We also calculate their asymptotic proportion in the set of the 'king permutations', i.e. the ones which satisfy only the first of the two conditions above. With this aim we define a new parameter on permutations, namely, the number of {\em cyclic bonds} which is a modification of the number of bonds. In addition, we present some results regarding the distribution of this parameter.

math.CO↗

arXiv2019

Separators - a new statistic for permutations

A digit $π_j$ in a permutation $π=[π_1,\ldots,π_n]\in S_n$ is defined to be a separator of $π$ if by omitting it from $π$ we get a new $2-$block. In this work we introduce a new statistic, the number of separators, on the symmetric group $S_n$ and calculate its distribution over $S_n$. We also provide some enumerative and asymptotic results regarding this statistic.

math.CO↗

arXiv2018

On two-sided gamma-positivity for simple permutations

Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin. We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.

math.CO↗

arXiv2014

A generalization of Tverberg's Theorem

The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d + 1)(r - 1) + 1 are known as Tverberg numbers. Reay asks the following question: if we add an additional parameter k (1 < k < r+1) what is the minimal number of points we need in order to guarantee that there exists an r partition of them such that any k of the r convex hulls intersect. This minimal number is denoted by T(d,r,k). Reay conjectured that T(d,r,k) = T(d,r) for all d,r and k. In this article we prove that this is true for the following cases: when k > [ (d+3)/2 ]-1 or when d < rk/(r-k)-1 and for the specific values d = 3; r = 4; k = 2 and d = 5; r = 3; k = 2.

math.CO↗

arXiv2014

Strong General Position

We say that a finite set S of points in R^d is in "strong general position" if for any collection {F_1,..., F_r} of r pairwise disjoint subsets of S (1 <= r <= |S|) we have: d-dim (the intersection of aff F_1,aff F_2,...,aff F_r) = min{d+1, (d-dim aff F_1)+(d-dim aff F_2)+...+(d-dim aff F_r)}. In this paper we reduce the set of conditions that one has to check in order to determine if S is in "strong general position".

math.CO↗