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"Permutations"
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Prime suspects : the anatomy of integers and permutations
Integers and permutations--two of the most basic mathematical objects--are born of different fields and analyzed with separate techniques. Yet when the Mathematical Sciences Investigation team of crack forensic mathematicians, led by Professor Gauss, begins its autopsies of the victims of two seemingly unrelated homicides, Arnie Integer and Daisy Permutation, they discover the most extraordinary similarities between the structures of each body.
Book
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.
Journal Article
Six new classes of permutation trinomials over F 3 3 k
Permutation trinomials over finite fields are of great interest for their simple algebraic forms and important applications in many areas of mathematics and engineering. In this paper, six new classes of permutation trinomials over F33k are presented based on the multivariate method. Their permutation properties are proved by using the resultant elimination method.
Journal Article
The permutation polynomial with the formula b(xq + ax + δ)i + c(xq + ax + δ)j + uxq + vx and their compositional inverses over q2
The field containing exactly q elements is referred to as q, where q is a power of a prime. The class of permutations polynomials (PP) with the formula f(x) = b(xq+ax+δ)i+c(xq+ax+δ)j+uxq+vx and its compositional inverse over Fq is examined in this work, where b, c ∈ q, a, δ, u and v ∈ q2 with a1+q = 1, (avq + u)(au − v) ≠ 0 and uq + v = a(u + avq).
Journal Article
Random Permutation Set
For exploring the meaning of the power set in evidence theory, a possible explanation of power set is proposed from the view of Pascal’s triangle and combinatorial number. Here comes the question: what would happen if the combinatorial number is replaced by permutation number? To address this issue, a new kind of set, named as random permutation set (RPS), is proposed in this paper, which consists of permutation event space (PES) and permutation mass function (PMF). The PES of a certain set considers all the permutation of that set. The elements of PES are called the permutation events. PMF describes the chance of a certain permutation event that would happen. Based on PES and PMF, RPS can be viewed as a permutation-based generalization of random finite set. Besides, the right intersection (RI) and left intersection (LI) of permutation events are presented. Based on RI and LI, the right orthogonal sum (ROS) and left orthogonal sum (LOS) of PMFs are proposed. In addition, numerical examples are shown to illustrate the proposed conceptions. The comparisons of probability theory, evidence theory, and RPS are discussed and summarized. Moreover, an RPS-based data fusion algorithm is proposed and applied in threat assessment. The experimental results show that the proposed RPS-based algorithm can reasonably and efficiently deal with uncertainty in threat assessment with respect to threat ranking and reliability ranking.
Journal Article
2×22×2 monotone grid classes are finitely based
In this note, we prove that all 2×22×2 monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain 2×22×2 (generalized) grid classes having two monotone cells in the same row.
Journal Article
Simulation of large-scale power database cache conflict handling based on frequent itemsets
This paper proposes a simulation method based on frequent itemsets to address the issue of imbalanced cache conflict handling caused by uneven data distribution in power databases. Firstly, the decision equilibrium coefficient is introduced to evaluate database load and identify conflict patterns; Secondly, frequent itemsets are mined, and conflict object sets are constructed by combining joint variables and cross entropy loss functions; Then, the conflicting objects are dispersed to different cache locations through permutation functions, and uniqueness checks are performed to ensure the validity of the mapping. The experimental results show that this method significantly improves the cache hit rate and achieves better cache load balancing performance compared to traditional strategies.
Journal Article
A NEW CHARACTERISATION FOR QUARTIC RESIDUACITY OF $\\mathbf {2}
Let p be a prime with
$p\\equiv 1\\pmod {4}$
. Gauss first proved that
$2$
is a quartic residue modulo p if and only if
$p=x^2+64y^2$
for some
$x,y\\in \\Bbb Z$
and various expressions for the quartic residue symbol
$(\\frac {2}{p})_4$
are known. We give a new characterisation via a permutation, the sign of which is determined by
$(\\frac {2}{p})_4$
. The permutation is induced by the rule
$x \\mapsto y-x$
on the
$(p-1)/4$
solutions
$(x,y)$
to
$x^2+y^2\\equiv 0 \\pmod {p}$
satisfying
$1\\leq x < y \\leq (p-1)/2$
.
Journal Article