Advances in Applied Mathematics最新文献

There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinations of products of theta functions, we present two strategies that will provide a unified treatment. Our approaches rely on studying the behavior of products of two theta series under the action of the huffing operator. For this purpose, some explicit criteria are given. We may use the presented methods to not only verify experimentally discovered coefficient-vanishing results, but also to produce a series of general phenomena.

For a subset D of boxes in an $n\times n$ square grid, let ${\chi }_D\left(x\right)$ denote the dual character of the flagged Weyl module associated to D. It is known that ${\chi }_D\left(x\right)$ specifies to a Schubert polynomial (resp., a key polynomial) in the case when D is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of ${\chi }_D\left(x\right)$. Mészáros, St. Dizier and Tanjaya conjectured that ${\chi }_D\left(x\right)$ attains the upper bound if and only if D avoids a certain single subdiagram. We provide a proof of this conjecture.

We provide a new algebraic solution procedure for the global positioning problem in n dimensions using m satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures. We fill a gap in the literature by giving a proof for the long-held belief that when $m\ge n+2$, the solution is unique for almost all user positions. Even better, when $m\ge 2n+2$, almost all satellite configurations will guarantee a unique solution for all user positions. Our uniqueness results provide a basis for predicting the behavior of numerical solutions, as ill-conditioning is expected near the threshold between areas of nonuniqueness and uniqueness. Some of our results are obtained using tools from algebraic geometry.

We came across an unexpected connection between a remarkable grammar of Dumont for the joint distribution of $(\mathrm{exc},\mathrm{fix})$ over $S_n$ and a beautiful theorem of Diaconis-Evans-Graham on successions and fixed points of permutations. With the grammar in hand, we demonstrate the advantage of the grammatical calculus in deriving the generating functions, where the constant property plays a substantial role. On the grounds of left successions of a permutation, we present a grammatical treatment of the joint distribution investigated by Roselle. Moreover, we obtain a left succession analogue of the Diaconis-Evans-Graham theorem, exemplifying the idea of a grammar assisted bijection. The grammatical labelings give rise to an equidistribution of $(\mathrm{jump},\mathrm{des})$ and $(\mathrm{exc},\mathrm{drop})$ restricted to the set of left successions and the set of fixed points, where jump is defined to be the number of ascents minus the number of left successions.

The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set ${\check{\Lambda }}^q$ of strict integer partitions (i.e., with unequal parts) into perfect q-th powers. A combinatorial link is provided via a suitable conditioning by fixing the partition weight (the sum of parts) and length (the number of parts), leading to uniform distribution on the corresponding subspaces of partitions. The Boltzmann measure is calibrated through the hyper-parameters $〈N〉$ and $〈M〉$ controlling the expected weight and length, respectively. We study “short” partitions, where the parameter $〈M〉$ is either fixed or grows slower than for typical partitions in ${\check{\Lambda }}^q$. For this model, we obtain a variety of limit theorems including the asymptotics of the cumulative cardinality in the case of fixed $〈M〉$ and a limit shape result in the case of slow growth of $〈M〉$. In both cases, we also characterize the joint distribution of the weight and length, as well as the growth of the smallest and largest parts. Using these results we construct suitable sampling algorithms and analyze their performance.

In the present paper, we introduce a new approach and use it to prove that the maximum number of triangles in a $C_5$-free graph on n vertices is at most $\frac{1}{2\sqrt{6}}(1+o(1\left)\right)n^{3/2}$, improving an estimate of Ergemlidze and Methuku [4]. We also show that the maximum size of an induced-$C_4$-free and $C_5$-free graph on n vertices is at most $\frac{1}{\sqrt{6}}(1+o(1\left)\right)n^{3/2}$, also improving an estimate of Ergemlidze and Methuku [4].

We determine the maximal non-central hyperplane sections of the $l_1^n$-ball if the fixed distance of the hyperplane to the origin is between $\frac{1}{\sqrt{3}}$ and $\frac{1}{\sqrt{2}}$. This adds to a result of Liu and Tkocz who considered the distance range between $\frac{1}{\sqrt{2}}$ and 1. For $n\ge 4$, the maximal sections are parallel to the $(n-1)$-dimensional coordinate planes. We also study non-central sections of the complex $l_{\infty }^2$-ball, where the formulas are more complicated than in the real case. Also, the extrema are partially different compared to the real case.

We introduce a kind of $(p,q,t)$-Catalan numbers of Type A by generalizing the J-type continued fraction formula, we prove that the corresponding expansions could be expressed by the polynomials counting permutations on $S_n\left(321\right)$ by various descent statistics. Moreover, we introduce a kind of $(p,q,t)$-Catalan numbers of Type B by generalizing the J-type continued fraction formula, we prove that the Taylor coefficients and their γ-coefficients could be expressed by the polynomials counting permutations on $S_n(3124,4123,3142,4132)$ by various descent statistics. Our methods include permutation enumeration techniques involving variations of bijections from permutation patterns to labeled Motzkin paths and modified Foata-Strehl action.

A strongly unimodal sequence of size n is a sequence of integers ${\left\{a_j\right\}}_{j=1}^s$ satisfying the following conditions:$0<a_1<a_2<\cdots <a_k>a_{k+1}>\cdots >a_s>0\text{and}{a}_1+a_2+\cdots +a_s=n,$ for a certain index k, and we usually define its rank as $s-2k+1$. Let $u(m,n)$ be the number of strongly unimodal sequences of size n with rank m, and the generating function for $u(m,n)$ is written as$U(z;q):=\sum _{m,n}u(m,n)z^m{q}^n.$ Recently, Chen and Garvan established some Hecke-type identities for the third order mock theta function $\psi \left(q\right)$ and $U\left(q\right)$, which are the specializations of $U(z;q)$, as advocated by $\psi \left(q\right)=U(\pm i;q)$ and $U\left(q\right)=U(1;q)$. Meanwhile, they inquired whether these Hecke-type identities could be proved via the Bailey pair machinery. In this paper, we not only answer the inquiry of Chen and Garvan in the affirmative, but offer more instances in a broader setting, with, for example, some classical third order mock theta functions due to Ramanujan involved. Furthermo