A universal cycle for the k-permutations of 〈n〉 = {1, 2, ..., n} is a circular string of length (n)k that contains each k-permutation exactly once as a substring. Jackson (Discrete Mathematics, 149 (1996) 123–129) proved their existence for all k ≤ n − 1. Knuth (The Art of Computer Programming, Volume 4, Fascicle 2, Addison-Wesley, 2005) pointed out the importance of the k = n − 1 case, where e...

Let K be a finite field of characteristic p . We study certain class functions → that agree with an F -affine function on each coset given additive subgroup W – we call them W-coset-wise show these form permutation group the structure imprimitive wreath product and characterize which are complete mappings As consequence, able to provide various new examples cycle types for instance, if > 2 , th...

A generalized Catalan matrix ( a n , k ) ≥ 0 is generated by two seed sequences s = 1 … and t 2 together with recurrence relation. By taking ℓ + we can interpret as the number of partial permutations, which are × -matrices zero rows at most one in each row or column. In this paper prove that fundamental statistics some set-valued on permutations also be defined encoded sequences. Results intere...

Let (G, D) be a permutation representation of a finite group G acting on a finite set D. The cycle index of this representation is a polynomial P(G, D; Xl ,..., x,~) in several variables xl ,..., x~ with rational numbers as coefficients (see [1]). The restriction, made in [1], that the representation (G, D) is faithful, is unnecessary and we put no restriction on (G, D) whatsoever. We replace e...

The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type ...

introduction: in this paper, a novel complexity measure is proposed to detect dynamical changes in nonlinear systems using ordinal pattern analysis of time series data taken from the system. epilepsy is considered as a dynamical change in nonlinear and complex brain system. the ability of the proposed measure for characterizing the normal and epileptic eeg signals when the signal is short or is...

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the “free cumulants” of the associated Young diagram. We present a positivity conjecture for their coefficients. The latter is stronger than the positivity conjecture due to Kerov and Biane. 1 Kerov polynomials 1.1 Characters A partition λ = (λ1, ......

Let C be a conjugation class of permutations of a finite field Fq. We consider the function NCðqÞ defined as the number of permutations in C for which the associated permutation polynomial has degree 5q 2. In 1969, Wells proved a formula for N1⁄23 ðqÞ where 1⁄2k denotes the conjugation class of k-cycles. We will prove formulas for N1⁄2k ðqÞ where k 1⁄4 4; 5; 6 and for the classes of permutation...

We show how to onstru t pseudo-random permutations that satisfy a ertain y le restri tion, for example that the permutation be y li ( onsisting of one y le ontaining all the elements) or an involution (a self-inverse permutation) with no xed points. The onstru tion an be based on any (unrestri ted) pseudo-random permutation. The resulting permutations are de ned su in tly and their evaluation a...