اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدInversion polynomials for 321-avoiding permutations
177
0
0.0
(
0
)
تأليف
Szu-En Cheng
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formulas for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.
قيم البحث
اقرأ أيضاً
This addendum contains results about the inversion number and major index polynomials for permutations avoiding 321 which did not fit well into the original paper. In particular, we consider symmetry, unimodality, behavior modulo 2, and signed enumeration.
An alternating permutation of length $n$ is a permutation $pi=pi_1 pi_2 ... pi_n$ such that $pi_1 < pi_2 > pi_3 < pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(sigma)$ be set of alternating permutations
in $A_n$ that avoid a pattern $sigma$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bona, Xu and Yan. In this paper, we prove the two relations $|A_{2n+1}(1243)|=|A_{2n+1}(2143)|$ and $|A_{2n}(4312)|=|A_{2n}(1234)|$ as conjectured by Lewis.
In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alter
nating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.
Babson and Steingr{i}msson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian s
tatistic in terms of generalized pattern functions, which is denoted $stat$. Recently, Amini investigated the equidistributions of these Mahonian statistics over sets of pattern avoiding permutations. Moreover, he posed several conjectures. In this paper, we construct a bijection from $S_n(213)$ to $S_n(231)$, which maps the statistic $(maj,stat)$ to the statistic $(stat,maj)$. This allows us to give solutions to some of Aminis conjectures.
Recently, Nunge studied Eulerian polynomials on segmented permutations, namely emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eule
rian polynomials and hence confirm Nunges conjecture. Our proof is based on Brandens stable multivariate Eulerian polynomials. By acting on Brandens polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunges conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
