Equidistribution and sign-balance on 132-avoiding permutations (original) (raw)

Equidistribution and Sign-Balance on 321-Avoiding Permutations

2003

Let TnT_nTn be the set of 321-avoiding permutations of order nnn. Two properties of TnT_nTn are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over TnT_nTn, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The sign-and-last-descent enumerators for T2nT_{2n}T2n and T2n+1T_{2n+1}T2n+1 are essentially equal to the last-descent enumerator for TnT_nTn. The proofs use a recursion formula for an appropriate multivariate generating function.

The Descent Statistic on 123-AVOIDING Permutations

2010

We exploit Krattenthaler's bijection between 123-avoi- ding permutations and Dyck paths to determine the Eulerian dis- tribution over the set Sn(123) of 123-avoiding permutations in Sn. In particular, we show that the descents of a permutation corre- spond to valleys and triple ascents of the associated Dyck path. We get the Eulerian numbers of Sn(123) by studying the joint distribution

A relation on 132-avoiding permutation patterns

Discrete Mathematics & Theoretical Computer Science, 2015

A permutation σσσ contains the permutation τττ if there is a subsequence of σσσ order isomorphic to τττ. A permutation σσσ is τττ-avoiding if it does not contain the permutation τττ. For any nnn, the popularity of a permutation τττ, denoted AAA$n$($τ$), is the number of copies of τττ contained in the set of all 132-avoiding permutations of length nnn. Rudolph conjectures that for permutations τττ and μμμ of the same length, AAA$n$($τ$) ≤ AAA$n$($μ$) for all nnn if and only if the spine structure of τττ is less than or equal to the spine structure of μμμ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of τττ is less than or equal to the spine structure of μμμ, then AAA$n$($τ$) ≤ AAA$n$($μ$) for all nnn. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.

The descent statistic over 123-avoiding permutations

2009

We exploit Krattenthaler’s bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set Sn(123) of 123-avoiding permutations in Sn. In particular, we show that the descents of a permutation correspond to valleys and triple falls of the associated Dyck path. We get the Eulerian numbers of Sn(123) by studying the joint distribution of these two statistics on Dyck paths.

Permutations avoiding 1324 and patterns in Łukasiewicz paths

The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.

Bijective Enumeration and Sign-Imbalance for Permutation Depth and Excedances

Electronic proceedings in theoretical computer science, 2024

We present a simplified variant of Biane's bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic socalled depth of a permutation. This generalizes a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the symmetric group S n of permutations. In terms of weighted Motzkin path, we establish an involution on S n that reverses the parities of depth and excedance numbers simultaneously, which proves that the numbers of permutations with even and odd depth (excedance numbers, respectively) are equal if n is even and differ by the tangent number if n is odd. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.

On the growth rate of 1324-avoiding permutations

2014

We give an improved algorithm for counting the number of 132413241324-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length nnn behaves as B\cdot \mu^n \cdot \mu_1^{n^{\sigma}} \cdot n^g.$$ We estimate mu=11.60pm0.01,\mu=11.60 \pm 0.01,mu=11.60pm0.01, sigma=1/2,\sigma=1/2,sigma=1/2, mu_1=0.0398pm0.0010,\mu_1 = 0.0398 \pm 0.0010,mu_1=0.0398pm0.0010, g=−1.1pm0.2g = -1.1 \pm 0.2g=−1.1pm0.2 and B=9.5pm1.0.B =9.5 \pm 1.0.B=9.5pm1.0.

A refined sign-balance of simsun permutations

European Journal of Combinatorics, 2014

We present a refined sign-balance result for simsun permutations. On the basis of our previously established bijection between simsun permutations and increasing 1-2 trees, we deduce the recurrence relation and exponential generating function for the sign-balance of simsun permutations of length n with k descents. For odd lengths, the distribution turns out to be (shifted) secondorder Eulerian numbers. For even lengths, the distribution forms a signed triangle whose row sums are all zeros. Meanwhile, we obtain two Pólya frequency sequences, one of which refines the double factorial of the odd numbers and the other, that of the even numbers.

The Location of the First Ascent in a 123-Avoiding Permutation

It is natural to ask, given a permutation with no three-term ascending subsequence, at what index the first ascent occurs. We shall show, using both a recursion and a bijection, that the number of 123-avoiding permutations at which the first ascent occurs at positions k,k+1k,k+1k,k+1 is given by the kkk-fold Catalan convolution Cn,kC_{n,k}Cn,k. For 1leklen1\le k\le n1leklen, Cn,kC_{n,k}Cn,k is also seen to enumerate the number of 123-avoiding permutations with nnn being in the kkkth position. Two interesting discrete probability distributions, related obliquely to the Poisson and geometric random variables, are derived as a result.

Counting permutations by alternating descents

The Electronic Journal of Combinatorics, 2014

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of David and Barton's generating function for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel.