A permutations representation that knows what "Eulerian" means

Mantaci, Roberto; Rakotondrajao, Fanja

doi:10.46298/dmtcs.271

Cited by 9 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Foata and Schützenberger [7] gave the fundamental work on these numbers. Mantaci and Rakotondrajao [15] gave a new combinatorial interpretation to the same. Many other references concerning Eulerian numbers can be found on the OEIS A000295.…”

Section: Introductionmentioning

confidence: 90%

Run Distribution Over Flattened Partitions

Nabawanda,

Rakotondrajao,

Bamunoba

2020

Preprint

View full text Add to dashboard Cite

The study of flattened partitions is an active area of current research. In this paper, our study unexpectedly leads us to the OEIS numbers A124324. We provide a new combinatorial interpretation of these numbers. A combinatorial bijection between flattened partitions over [n + 1] and the partitions of [n] is also given in a separate section. We introduce the numbers f n,k which count the number of flattened partitions over [n] having k runs. We give recurrence relations defining them, as well as their exponential generating function in differential form. It should be appreciated if its closed form is established. We extend the results to flattened partitions where the first s integers belong to different runs. Combinatorial proofs are given.

show abstract

Section: Introductionmentioning

confidence: 90%

Run Distribution Over Flattened Partitions

Nabawanda,

Rakotondrajao,

Bamunoba

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…We use the subscript I to emphasize that this is a statistic for inversion sequences which is different from the ascent statistic for permutations used earlier in the paper. Mantaci and Rakotondrajao (2001) also studied this representation of permutations under the name "subexceedant functions". They considered the statistic that counts that distinct entries in e ∈ I n , dst(π) = |{e i : 1 ≤ i ≤ n}|.…”

Section: Connection To Inversion Sequencesmentioning

confidence: 99%

Some remarks on the joint distribution of descents and inverse descents

Visontai¹

2012

Preprint

View full text Add to dashboard Cite

We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture.We examine generalizations of the conjecture and obtain a type B analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution in terms of statistics on inversion sequences.Conjecture 1.1 (Gessel). For all n ≥ 1,where γ n,i,j are nonnegative integers for all i, j ∈ N.

show abstract

“…The Lehmer code of a permutation f is defined as the number of indices j such that 1 ≤ j < i and f (j) < f (i) [17].…”

Section: Ranking and Unranking Permutations Of Given Size With Lehmer...mentioning

confidence: 99%

Ranking and Unranking of Hereditarily Finite Functions and Permutations

Tarau

2008

Preprint

View full text Add to dashboard Cite

Prolog's ability to return multiple answers on backtracking provides an elegant mechanism to derive reversible encodings of combinatorial objects as Natural Numbers i.e. ranking and unranking functions. Starting from a generalization of Ackerman's encoding of Hereditarily Finite Sets with Urelements and a novel tupling/untupling operation, we derive encodings for Finite Functions and use them as building blocks for an executable theory of Hereditarily Finite Functions. The more difficult problem of ranking and unranking Hereditarily Finite Permutations is then tackled using Lehmer codes and factoradics. The paper is organized as a self-contained literate Prolog program available at http://logic.csci.unt.edu/tarau/research/2008/pHFF.zip.

show abstract

A permutations representation that knows what "Eulerian" means

Cited by 9 publications

References 6 publications

Run Distribution Over Flattened Partitions

Run Distribution Over Flattened Partitions

Some remarks on the joint distribution of descents and inverse descents

Ranking and Unranking of Hereditarily Finite Functions and Permutations

Contact Info

Product

Resources

About