Prefix Exchanging and Pattern Avoidance by Involutions

Permutation patterns and statistics

In a recent paper, Backelin, West and Xin describe a map φ* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The resulting permutation φ*(σ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ* commutes with taking the inverse of a permutation. In the BWX paper, the definition of φ* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map φ* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T′ be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group $${\cal S}$$ n that avoid T equals the number of permutations of $${\cal S}$$ n that avoid T′. Our commutation result, generalized to Ferrers boards, implies that the number of involutions of $${\cal S}$$ n that avoid T is equal to the number of involutions of $${\cal S}$$ n avoiding T′, as recently conjectured by Jaggard.

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n?The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multi-normal of order \(1/\sqrt n \), while the orthogonal component is smaller.Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to Sk-representations.Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

We consider the problem of enumerating permutations in the symmetric group on n elements which avoid a given set of consecutive patterns S, and in particular computing asymptotics as n tends to infinity. We develop a general method which solves this enumeration problem using the spectral theory of integral operators onL2([0,1]m), where the patterns in S have length m+1.Kre\u{\i}n and Rutman’s generalization of the Perron–Frobenius theory of non-negative matrices plays a central role. Our methods give detailed asymptotic expansions and allow for explicit computation of leading terms in many cases.As a corollary to our results,we settle a conjecture of Warlimont on asymptotics for the number of permutations avoiding a consecutive pattern.

Let X 1, . . . ,X N denote N independent d-dimensional Levy processes, and consider the N-parameter random field $$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$ First we demonstrate that for all nonrandom Borel sets $${F\subseteq{{\bf R}^d}}$$ , the Minkowski sum $${\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}$$ , of the range $${\mathfrak{X}({{\bf R}^{N}_{+}})}$$ of $${\mathfrak{X}}$$ with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Levy processes. Presently, we highlight a few new consequences.

For a symmetric group G:=symn>G:=symnG:=symn and a conjugacy class X>XX of involutions in G>GG‎, ‎it is known that if the class of involutions does not have a unique fixed point‎, ‎then‎ - ‎with a few small exceptions‎ - ‎given two elements a,x∈X>a,x∈Xa,x∈X‎, ‎either ⟨a,x⟩>⟨a,x⟩⟨a,x⟩ is isomorphic to the dihedral group D8>D8D8‎, ‎or there is a further element y∈X>y∈Xy∈X such that ⟨a,y⟩≅⟨x,y⟩≅D8>⟨a,y⟩≅⟨x,y⟩≅D8⟨a,y⟩≅⟨x,y⟩≅D8 (P‎. ‎Rowley and D‎. ‎Ward‎, ‎On π>ππ-Product Involution Graphs in Symmetric‎ ‎Groups‎. ‎MIMS ePrint‎, ‎2014)‎. ‎One natural generalisation of this to p>pp-elements is to consider when two conjugate p>pp-elements generate a wreath product of two cyclic groups of order p>pp‎. ‎In this paper we give necessary and sufficient conditions for this in the case that our p>pp-elements have full support‎. ‎These conditions relate to given matrices that are of circulant or permutation type‎, ‎and corresponding polynomials that represent these matrices‎. ‎We also consider the case that the elements do not have full support‎, ‎and see why generalising our results to such elements would not be a natural generalisation‎.

We investigate subadditivity of Ptak function as a sufficient and necessary condition for symmetry in the class of involutive spectrally bounded algebras. Among others it is proved an analogue of Raikov's criterion for complete lmc*-algebras in which the hermitian elements have finite spectral radius.

Sorted and/or sortable permutations

For a function f \colon \{0,1\}^{n} \to \mathbb{R} with Fourier expansion f=\sum_{S \subset \{1,\ldots,n\}}\hat f(S)\chi_{S} , the hypercontractive inequality for the noise operator allows bounding norms of T_{\rho} f = \sum_{S} \rho^{|S|}\hat f(S)\chi_{S} in terms of norms of f . If f is Boolean-valued, the level- d inequality allows bounding the norm of f^{=d}=\sum_{|S|=d}\hat f(S)\chi_{S} in terms of \mathbb{E}[f] . These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube \{0,1\}^{n} is endowed with the uniform measure, it is easy to show that they do not hold for more general discrete product spaces, and finding a ‘natural’ generalization was a long-standing open problem. Keevash, Lifshitz, Long, and Minzer [J. Amer. Math. Soc. 37, 245–279 (2024)] obtained a hypercontractive inequality for general discrete product spaces, that holds for functions which are ‘global’ – namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group S_{n} and to other settings where sharpness of the bound is crucial. Also, no sharp level- d inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level- d inequality for global functions over discrete product spaces. Our inequalities open the way for diverse applications to extremal set theory, group theory, theoretical computer science, and number theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications that were obtained using our results. Those contain applications to the study of functions over the symmetric group S_{n} – including hypercontractivity and level- d inequalities, character bounds, variants of Roth’s theorem and of Bogolyubov’s lemma, and diameter bounds, as well as an application to the Furstenberg–Sárközy problem on the maximal size of a subset of \{1,\ldots,n\} which does not contain two elements that differ by a perfect square.

We derive basis-independent, necessary and sufficient conditions for the custodial symmetry in N-Higgs-doublet models (NHDM) for N ≥ 3, and apply them on some 3HDM examples.

Previous analyses have shown that if a point x is to be a core of a majority rule voting game in Euclidean space, when preferences are smooth, then the utility gradients must satisfy certain restrictive symmetry conditions. In this paper these results are generalized to the case of an arbitrary voting rule, and necessary and sufficient conditions, expressed in terms of pivotal coalitions, are obtained.

Spread mooring systems (SMS) are labeled as general when they are not restricted by conditions of symmetry. The six necessary and sufficient conditions for stability of general SMS are derived analytically. The boundaries where static and dynamic loss of stability occur also are derived in terms of the system eigenvalues, thus providing analytical means for defining the morphogenesis that occurs when a bifurcation boundary is crossed. The equations derived in this paper provide analytical expressions of elementary singularities and routes to chaos for general mooring system configurations. Catastrophe sets are generated first by the derived expressions and then numerically using nonlinear dynamics and codimension-one and -two bifurcation theory; agreement is excellent. The mathematical model consists of the nonlinear, third-order maneuvering equations without memory of the horizontal plane, slow-motion dynamics—surge, sway, and yaw—of a vessel moored to several terminals. Mooring lines can be modeled by synthetic nylon ropes, chains, or steel cables. External excitation consists of time-independent current, wind, and mean wave drift forces. The analytical expressions derived in this paper apply to nylon ropes and current excitation. Expressions for other combinations of lines and excitation can be derived.

This paper presents a systematic isotropy analysis of a caster wheeled omnidirectional mobile robot (COMR) with three active caster wheels. Unlike previous analysis, no assumption is made on the relative scale of the steering link offset and the wheel radius. First, with the characteristic length introduced, the kinematic model of a COMR is obtained based on the orthogonal decomposition of the wheel velocities. Second, the necessary and sufficient isotropy conditions are examined to categorize three different groups to be handled in a similar way. Third, the isotropy conditions are further explored to identify four different sets of all possible isotropic configurations. Fourth, the characteristic lengths required for the isotropy of a COMR are obtained in a closed-form. Finally, the local and the global isotropy indices are used to determine the optimal design parameters.KeywordsOmnidirectional mobile robotCaster wheelSteering link offsetIsotropy analysis

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Sufficient reduction conditions for partial differential equations possessing nontrivial conditional symmetry are established. The results obtained generalize the classical reduction conditions of differential equations by means of group-invariant solutions. A number of examples illustrating the reduction in the number of independent and dependent variables of systems of partial differential equations are considered.