scholarly journals On the Möbius Function of Permutations with One Descent

10.37236/3559 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Jason P. Smith
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Pattern Containment

The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the Möbius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the Möbius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the Möbius function is unbounded on the poset of all permutations. We show that the Möbius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the Möbius function on some other intervals of permutations with at most one descent.

scholarly journals A Formula for the Möbius Function of the Permutation Poset Based on a Topological Decomposition

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jason P Smith
Keyword(s):  
Significant Proportion ◽  
Möbius Function ◽  
Permutation Patterns ◽  
Mobius Function ◽  
International Audience ◽  
Definition Of ◽  
Object Of Study ◽  
Pattern Containment ◽  
Exponential Complexity ◽  
Exponential Part

International audience The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.

scholarly journals The Möbius Function of the Consecutive Pattern Poset

10.37236/633 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Antonio Bernini ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Möbius Function ◽  
Mobius Function ◽  
Permutation Pattern ◽  
Pattern Containment

An occurrence of a consecutive permutation pattern $p$ in a permutation $\pi$ is a segment of consecutive letters of $\pi$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the Möbius function of intervals in this poset. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the Möbius function. In particular, we show that the Möbius function only takes the values $-1$, $0$ and $1$.

scholarly journals The Möbius function of separable permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Vít Jelínek ◽  
Eva Jelínková ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Computationally Efficient ◽  
Mobius Function ◽  
Nous Montrons ◽  
International Audience ◽  
Separable Permutations ◽  
Pattern Containment

International audience A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. Using the notion of separating tree, we give a computationally efficient formula for the Möbius function of an interval $(q,p)$ in the poset of separable permutations ordered by pattern containment. A consequence of the formula is that the Möbius function of such an interval $(q,p)$ is bounded by the number of occurrences of $q$ as a pattern in $p$. The formula also implies that for any separable permutation $p$ the Möbius function of $(1,p)$ is either 0, 1 or -1. Une permutation est séparable si elle peut être générée á partir de la permutation 1 par des sommes directes et des sommes indirectes, ou de façon équivalente, si elle évite les motifs 2413 et 3142. En utilisant le concept de l'arbre séparant, nous donnons une formule pour le calcul efficace de la fonction de Möbius d'un intervalle de $(q, p)$ dans l'ensemble partiellement ordonné des permutations séparables. Une conséquence est que la fonction de Möbius de $(q,p)$ pour $q$ et $p$ séparables est bornée par le nombre d'occurrences de $q$ comme un motif en $p$. Nous montrons aussi que pour une permutation $p$ séparable, la fonction de Möbius de $(1,p)$ est soit 0, 1 ou -1.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Does the Möbius Function Determine Multiplicative Arithmetic?

1995 ◽  
Vol 102 (4) ◽  
pp. 354-356
Author(s):  
D. Flath ◽  
A. Zulauf
Keyword(s):  
Möbius Function ◽  
Mobius Function ◽  
Multiplicative Arithmetic

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

scholarly journals From explicit estimates for primes to explicit estimates for the Möbius function

2013 ◽  
Vol 157 (4) ◽  
pp. 365-379 ◽  
Author(s):  
Olivier Ramaré
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

scholarly journals A natural proof of the cyclotomic identity

1990 ◽  
Vol 42 (2) ◽  
pp. 185-189 ◽  
Author(s):  
D.E. Taylor
Keyword(s):  
Natural Isomorphism ◽  
Möbius Function ◽  
Mobius Function ◽  
Image Position

The cyclotomic identitywhere and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

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