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This document is from the College Mathematics Journal and contains a section with problems and solutions for college mathematics students and teachers. The editors encourage active participation by submitting solutions and proposing new and interesting problems. The document provides instructions for submitting problems and solutions, including the accepted file formats. It also includes a list of proposed problems and their corresponding solutions. Additionally, the document includes an editorial comment regarding a previously published incorrect solution and provides the intended solution for that problem. [Extracted from the article]

In this paper, we solve an open problem considered by Steven Finch (Central Binomial Coefficients, 2007, Available from: , p. 5), as far back as 2007, concerning the calculation of a series involving the central binomial coefficient. [ABSTRACT FROM AUTHOR]

We produce congruences modulo a prime p > 3 for sums ∑k3k k xk over ranges 0 ≤ k < q and 0 ≤ k < q/3, where q is a power of p. Here x equals either c2/(1 − c)3, or 4s2/(27(s2 − 1)), where c and s are indeterminates. In the former case, we deal more generally with shifted binomial coefficients 3k+e k. Our method derives such congruences directly from closed forms for the corresponding series. [ABSTRACT FROM AUTHOR]

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑ k = 0 , k ≠ n m + n (− 1) k n − k m + n k = (− 1) n m + n n (H m − H n) , where n , m are non-negative integers and H n is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums. [ABSTRACT FROM AUTHOR]

The given text discusses the concept of representative points in a graph and how they can be used to define a function. It explains that by choosing three rational points, one inside each region of the graph, a function can be created. The text aims to show that this function is injective, meaning that it is one-to-one. It provides a diagram and references a related problem about the possibility of having uncountably many disjoint figure 8s in the plane. The text concludes by mentioning that the result discussed is due to R. L. Moore. [Extracted from the article]

In this paper we consider a number triangle known as the Rascal triangle and place it within an infinite sequence of number triangles we call Iterated Rascal triangles. Initially, the name "Rascal triangle" was a clever play of words. However, we show a surprising and direct connection between Iterated Rascal triangles and Pascal's triangle. We prove an unexpected and elegant recursive formula generates the entries in any Iterated Rascal triangle. We establish formulas, generating functions, and combinatorial identities for them. From these principals we prove a version of Lucas' Theorem for the Iterated Rascal triangles. Moreover, we consider an identity from the Rascal triangle that has a well-known analogous identity in Pascal's triangle and extend it to all Iterated Rascal triangles. Although many can be made, we conclude with one conjecture. [ABSTRACT FROM AUTHOR]

The Boolean map χ n : F 2 n → F 2 n , x ↦ y defined by y i = x i + (x i + 1 + 1) x i + 2 (where i ∈ Z / n Z ) is used in various permutations that are part of cryptographic schemes, e.g., Keccak-f (the SHA-3-permutation), ASCON (the winner of the NIST Lightweight competition), Xoodoo, Rasta and Subterranean (2.0). In this paper, we study various algebraic properties of this map. We consider χ n (through vectorial isomorphism) as a univariate polynomial. We show that it is a power function if and only if n = 1 , 3 . We furthermore compute bounds on the sparsity and degree of these univariate polynomials, and the number of different univariate representations. Secondly, we compute the number of monomials of given degree in the inverse of χ n (if it exists). This number coincides with binomial coefficients. Lastly, we consider χ n as a polynomial map, to study whether the same rule ( y i = x i + (x i + 1 + 1) x i + 2 ) gives a bijection on field extensions of F 2 . We show that this is not the case for extensions whose degree is divisible by two or three. Based on these results, we conjecture that this rule does not give a bijection on any extension field of F 2 . [ABSTRACT FROM AUTHOR]

Classical hypergeometric series are reformulated as analytic functions of their parameters (in both the numerator and the denominator). Then, the coefficient extraction method is applied to examine hypergeometric series transformations. Several new closed form evaluations are established for harmonic series containing multinomial coefficient 4 n n , n , n , n and central binomial coefficient 2 n n . These results exclusively concern the alternating series of convergence rate " − 1 / 4 ". [ABSTRACT FROM AUTHOR]

To capture the higher-order autocorrelation structure for finite-range integer-valued time series of counts, and to consider the driving effect of covariates on the underlying process, this paper introduces a pth-order random coefficients mixed binomial autoregressive process with explanatory variables. The basic probabilistic and statistical properties of the model are discussed. Conditional least squares and conditional maximum likelihood estimators, as well as their asymptotic properties of the estimators are obtained. Moreover, the existence test of explanatory variables are well addressed using a Wald-type test. Forecasting problem is also considered. Finally, some numerical results of the estimators and a real data example are presented to show the performance of the proposed model. [ABSTRACT FROM AUTHOR]

Let p1,...,pr$p_1,\ldots,p_r$ be pairwise distinct primes. From a theorem of Kummer, each prime pj$p_j$ can divide 2nn${2n \atopwithdelims ()n}$ at most 1+logn/logpj$1 + \log n / \log p_j$ times. We show that, for all ε>0$\varepsilon > 0$, if p1,...,pr$p_1,\ldots,p_r$ are sufficiently large in terms of r$r$ and ε$\varepsilon$, then there exist infinitely many positive integers n$n$ such that each pj$p_j$ divides 2nn${2n \atopwithdelims ()n}$ at most ε(1+logn/logpj)$\varepsilon (1 + \log n / \log p_j)$ times. We connect this result to a famous conjecture by Graham on whether there are infinitely many integers n$n$ such that 2nn${2n \atopwithdelims ()n}$ is coprime to 105. [ABSTRACT FROM AUTHOR]

Three classes of infinite series containing binomial coefficient (3nn), harmonic-like numbers, and an independent variable “y” are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert’s series and manipulating the cubic transformations for the 3F2-series through the “coefficient extraction” method. [ABSTRACT FROM AUTHOR]

Yamamoto’s integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto’s integral. We determine the value of infinite series of the form: ∑ m1,...,mn,ℓ1,...,ℓk≥1 H(a1)m1 · · · H(an)mn/mb11 · · · mbnn ℓc11 · · · ℓckk (m1+···+mn+ℓ1+···+ℓkℓk), in terms of a finite sum of multiple zeta values, for positive integers a1, . . ., an, b1, . . ., bn, c1, . . ., ck. [ABSTRACT FROM AUTHOR]

The dynamic of different grass strips width effects on runoff-sediment reduction benefit are not clearly understood. Thus, we aim to determine the optimum grass strips width on runoff-sediment reduction benefits on inceptisols slopes. In this work, using a soil flume as a carrier, we analyzed the influences of flow rate and grass strips width on runoff and sediment yield, as well as determined the optimum grass strips width on runoff and sediment reduction benefit through simulated scouring experiments under different flow rates (4, 6 and 8 L/min) and grass strips widths (0.50, 0.75 and 1.00 m). The results showed that 1) the runoff coefficient increases binomial with increasing scouring time, and the runoff and sediment yield of grass strips were much lower than those of bare soil under a given flow rate. 2) Compared to the bare slope, the mean runoff and sediment reduction benefits of the average grass strips width were 27.20% and 43.66%, respectively. 3) The 1.00 m grass strips width had a great runoff-sediment benefit for controlling the soil erosion, and the runoff-sediment reduction benefit was more sensitive to changes in grass strips width than to flow rate. Therefore, we can conclude that the 1.00 m grass strips width was the optimal grass strips width for reducing the runoff and sediment yield. These results have theoretical and practical significance for preventing water and soil loss on purple soil sloping farmland. [ABSTRACT FROM AUTHOR]

Construction and enumeration of the class of balanced rotation symmetric Boolean functions are important research areas in cryptography. The counting results of this class of functions are available only for n = p , n = 2 p , and n = p q (where p and q are distinct primes). An explicit formula for counting balanced rotation symmetric Boolean functions for general n has been an open problem for the last few decades. This paper solves the above open problem using the concept of k -partition of multisets with equal sums. We extend this approach to construct and enumerate the balanced rotation symmetric functions over F p. [ABSTRACT FROM AUTHOR]

Five classes of multiple sums about circular products of binomial coefficients are investigated. Their generating functions are explicitly expressed as rational functions, with coefficients being Fibonacci and Lucas numbers. This is fulfilled by a novel approach called "recursive construction", which also permits us to establish, for the circular sums, both generating functions and recurrence relations through the Lagrange expansion formula. [ABSTRACT FROM AUTHOR]

We introduce the multi-Gaussian polynomial G k (M , N) , a multi-partition analogue of the Gaussian polynomial (also known as q -binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of G k (M , N). We also derive a Sylvester-type identity and its application. [ABSTRACT FROM AUTHOR]

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset S of Z. Bhargava's factorials k ! S are invariants, constructed using the notion of p -orderings of S where p is a prime. This paper defines b -orderings of any nonempty subset S of Z for all integers b ≥ 2 , as well as "extreme" cases b = 1 and b = 0. It defines generalized factorials k ! S , T and generalized binomial coefficients ( k + ℓ k ) S , T as nonnegative integers, for all nonempty S and allowing only b in T ⊆ N. It computes b -ordering invariants when S is Z and when S is the set of all primes. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and m and s positive integers, with m even. Let further F p m be the finite field of p m elements and R = F p m + u F p m ( u 2 = 0 ). Then R is a finite chain ring of p 2 m elements, and there is a Gray map from R N onto F p m 2 N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over F p m by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length p s over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length p s over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Pascal matrices presented in [9] as a functional for several variables. The entries of these matrices are defined on the set R ⊆ Z ≥ 0 n with multidimensional binomial coefficients along with several variables. In the following, we will investigate the algebraic properties of these matrices in terms of products, powers, and inverses of the matrices and also some factorization formulas using this method. [ABSTRACT FROM AUTHOR]

In this paper, we mainly prove some congruences involving binomial coefficients and binary quadratic forms. One such example is the following: Let p b be a prime such that p = x 2 + 2 y 2 ≡ 1 (mod 8) . Then, p ∑ k = 0 p - 1 2 k k 2 (8 k + 1) 16 k ≡ 3 p ∑ k = 0 p - 1 2 k k 2 (8 k + 3) 16 k ≡ 4 x 2 - 2 p - p 2 4 x 2 (mod p 3). [ABSTRACT FROM AUTHOR]

By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five conjectured recently by Z.-W. Sun ['Series with summands involving harmonic numbers', Preprint, 2023, arXiv:2210.07238v7]. This is realised by 'the coefficient extraction method' implemented by Mathematica commands. [ABSTRACT FROM AUTHOR]

The Ehrhart polynomial ehrP(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f*-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehrP(n) with respect to the binomial coefficient basis { n − 1 0 , n − 1 1 , ... , n − 1 d } , where d = dim P. Similarly to h/h*-vectors, the f*-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f*-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f*-coefficients increases and the last quarter decreases. Even though f*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h*-vector, there is a polytope with the same h*-vector whose f*-vector is unimodal. [ABSTRACT FROM AUTHOR]

A special case of an Entry in Part II of Ramanujan's Notebooks is such that \[ 1+\frac{1}{5} \left(\frac{1}{2} \right)^{2} + \frac{1}{9} \left(\frac{1\cdot 3}{2 \cdot 4} \right)^{2} + \cdots = \frac{\Gamma^4 \left(\frac{1}{4}\right)}{16\pi ^2}. \] 1 + 1 5 (1 2) 2 + 1 9 (1 ⋅ 3 2 ⋅ 4) 2 + ⋯ = Γ 4 (1 4) 16 π 2 . This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: \[ 1 + \frac{1}{5} \left(\frac{1}{2} \right)^{4} + \frac{1}{9} \left(\frac{1\cdot 3}{2\cdot 4} \right)^{4}+\cdots =\frac{\Gamma^8 \left(\frac{1}{4}\right)}{96\pi ^5}. \] 1 + 1 5 (1 2) 4 + 1 9 (1 ⋅ 3 2 ⋅ 4) 4 + ⋯ = Γ 8 (1 4) 96 π 5 . Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov. [ABSTRACT FROM AUTHOR]

In this work, by means of the generating function method and the De Moivre's formula, we derive some interesting combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers. One of them confirms the formula proposed recently by Svinin (2022). [ABSTRACT FROM AUTHOR]

The object of research is the method and algorithm of arithmetic addition of binomial numbers generated by binary binomial counting systems. The lack of binomial arithmetic, in particular the operation of adding binary binomial numbers, in a certain way prevents their introduction into information systems and the construction of information and communication technologies based on them for combinatorial optimization, generation of combinatorial objects, data compression and encryption. In the framework of the proposed approach, instead of operating with binomial coefficients, only operations with their upper and lower parameters are carried out. At the same time, the weighting coefficients of binary binomial numbers, which are added to each other, are represented in the form of two-component tuples. Taking this into account, this paper presents an algorithm for binomial arithmetic addition using dynamic arrays. The main idea, which is included in the structure of the algorithm of binomial arithmetic addition based on dynamic arrays, is that the transition from a two-dimensional model of summation to a one-dimensional one is carried out. At the same time, only available, existing binomial coefficients are placed in the dynamic array. Accordingly, the search for binomial coefficients equal to or greater than the quantitative equivalent takes place in much smaller areas. In comparison with the algorithm based on matrix models, this quite significantly reduces the amount of time spent when performing the summation operation, and also reduces the requirements for the amount of memory required for placing two-component tuples of the assembly array. In the course of the research, a several-fold decrease in the number of machine cycles required to search for the necessary elements in the dynamic array was practically confirmed. This leads to an increase in the performance of the presented algorithm of binomial arithmetic addition based on dynamic arrays. In turn, this leads to the acceleration of solving information tasks of combinatorial optimization, generation of combinatorial objects, data compression and encryption, for the solution of which the operation of adding binary binomial numbers is used. [ABSTRACT FROM AUTHOR]

Paratemporal methods based on tree expansion have proven to be effective in efficiently generating the trajectories of stochastic systems. However, combinatorial explosion of branching arising from multiple choice points presents a major hurdle that must be overcome to implement such techniques. In this paper, we tackle this scalability problem by developing a systems theory-based framework covering both conventional and proposed tree expansion algorithms for speeding up discrete event system stochastic simulations while preserving the desired accuracy. An example is discussed to illustrate the tree expansion framework in which a discrete event system specification (DEVS) Markov stochastic model takes the form of a tree isomorphic to a free monoid over the branching alphabet. We derive the computation times for baseline, non-merging, and merging tree expansion algorithms to compute the distribution of output values at any given depth. The results show the remarkable reduction from exponential to polynomial dependence on depth effectuated by node merging. We relate these results to the similarly reduced computation time of binomial coefficients underlying Pascal's triangle. Finally, we discuss the application of tree expansion to estimating temporal distributions in stochastic simulations involving serial and parallel compositions with potential real-world use cases. [ABSTRACT FROM AUTHOR]

Let |$\Phi $| be a root system. Postnikov introduced and studied the mixed |$\Phi $| -Eulerian numbers. These numbers indicate the mixed volumes of |$\Phi $| -hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget–Spink–Tseng gave a simple computation for the mixed |$\Phi $| -Eulerian numbers when |$\Phi $| is of type |$A$|⁠. In this paper, we connect a relation between mixed |$\Phi $| -Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget–Spink–Tseng in terms of left–right diagrams that were introduced by Abe–Horiguchi–Kuwata–Zeng for the purpose of Peterson Schubert calculus. We also derive a simple computation for the mixed |$\Phi $| -Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus. [ABSTRACT FROM AUTHOR]

In the present work, utilizing the known series, new series involving reciprocals of binomial coefficients, alternating positive, and negative binomial coefficients are constructed. Further, several new series of reciprocals of binomial coefficients with two odd terms in the denominator are obtained. In the end, we use these to establish the closed form evaluations of hypergeometric functions for the argument 1/16. [ABSTRACT FROM AUTHOR]

In this paper, we consider the sum of divisors d of n such that n / d is a power of 2 and derive a new decomposition for the number of plane partitions of n in terms of binomial coefficients as a sum over partitions of n. In this context, we introduce a new combinatorial interpretation of the number of plane partitions of n. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a fully modified (p,q)-poly-Euler polynomials and numbers of the first type. We investigate some properties that is related with (p; q)-Gaussian binomials coefficients. We also construct (p; q)- analogue of the Stirling numbers of the second kind and fully modified (p,q)-poly-Euler polynomials and numbers of the first type with two variables. [ABSTRACT FROM AUTHOR]

We present some new linear, quadratic, cubic, and quartic binomial Fibonacci, Lucas, and Fibonacci--Lucas summation identities. [ABSTRACT FROM AUTHOR]

The Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every kth person will be executed going around the circle until only one remains, find the position L (n , k) in which you should stand in order to be the last survivor. Let J n be the order in which the first person is executed on counting when k = 2 . In this paper, we consider the sequence (J n) n ⩾ 1 in order to introduce new expressions for the generating functions of the number of strict plane partitions and the number of symmetric plane partitions. This approach allows us to express the number of strict plane partitions of n and the number of symmetric plane partitions of n as sums over partitions of n in terms of binomial coefficients involving J n . Also, we introduce interpretations for the strict plane partitions and the symmetric plane partitions in terms of colored partitions. Connections between the sum of the divisors' functions and J n are provided in this context. [ABSTRACT FROM AUTHOR]

By applying the "coefficient extraction method" to hypergeometric series, we establish several remarkable infinite series identities about harmonic numbers and five binomial coefficients, including three conjectured by Z.-W. Sun. [ABSTRACT FROM AUTHOR]

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerategammafunction, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums. [ABSTRACT FROM AUTHOR]

In the article, by virtue of Maclaurin's expansions of the arcsine function and its square and cubic, the authors (1) give a short proof of a sum formula of a Maclaurin's series with coefficients containing reciprocals of the Catalan numbers; (2) establish four sum formulas for finite sums containing the ratio or product of two central binomial coefficients or the Catalan numbers. The instant proof simplifies discussions in the journal papers: College Math. J. 43 (2012), no. 2, 141-146; Amer. Math. Monthly 121 (2014), no. 3, 267-267; Amer. Math. Monthly 123 (2016), no. 4, 405-406; Elem. Math. 71 (2016), no. 3, 109-121; and Mathematics 5 (2017), no. 3, Article 40, 31 pages. [ABSTRACT FROM AUTHOR]

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub (Ann Comb, 23:725–748, 2019) showed that these results can be extended to the q-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining the arithmetic properties of the q-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser (Sém Lothar Combin, 81:24, 2020), to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, q- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the λ-Stirling numbers of both kinds, which are λ-analogues of Stirling numbers of both kinds. These numbers have nice combinatorial interpretations when λ are positive integers. If λ = 1, then the λ-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the λ-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the λ-Stirling numbers of both kinds are obtained for the λ-analogues of r-Stirling numbers of both kinds, which are generalizations of those numbers. [ABSTRACT FROM AUTHOR]

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. [ABSTRACT FROM AUTHOR]

The attractive properties of the hypercube graph such as its diameter, good connectivity, and symmetry have made it a popular topology for the design of multi-computer interconnection networks. Efforts to improve some of these properties have led to the evolution of hypercube variants. Let c be the proper coloring of graph G, where the neighboring vertices will get individual colors. Coloring c is irregular if distinct vertices have distinct color codes and the least number of colors that ought to receive an irregular coloring is the irregular chromatic number, χir (G). In this paper, we discuss the irregular coloring and find the irregular chromatic number for the hypercube graph Qn and some of its variants using binomial coefficients for the Locally twisted cube graph LTQn, Crossed cube graph CQn and two types of Fractal cubic network graph FCNG1 (k) and FCNG2 (k). [ABSTRACT FROM AUTHOR]

The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For r ∈ { 0 , 1 } , let Q L r (n) be the number of n-color partitions of n into distinct parts which have a number of parts congruent to r modulo 2. In this paper, we consider specializations of complete and elementary symmetric functions in order to establish two kinds of formulas for Q L 0 (n) ± Q L 1 (n) as sums over partitions of n in terms of binomial coefficients. The first kind of formulas only involve partitions in which the parts of size n appear at most n times, while the second kind of formulas involve unrestricted partitions. Similar results are obtained for the first differences of Q L 0 (n) ± Q L 1 (n) and the partial sums of Q L 0 (n) ± Q L 1 (n) . [ABSTRACT FROM AUTHOR]

Let a = (a 1 , a 2 , ... , a n) and e = (e 1 , e 2 , ... , e n) be real sequences. Denote by M e → a the (n + 1) × (n + 1) matrix whose (m , k) entry (m , k ∈ { 0 , ... , n }) is the coefficient of the polynomial (x − a 1) ⋯ (x − a k) in the expansion of (x − e 1) ⋯ (x − e m) as a linear combination of the polynomials 1 , x − a 1 , ... , (x − a 1) ⋯ (x − a m). By appropriate choice of a and e the matrix M e → a can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers. In all four of these examples, M e → a enjoys the property of total non-negativity — the determinants of all its square submatrices are non-negative. This leads to a natural question: when, in general, is M e → a totally non-negative? Galvin and Pacurar found a simple condition on e that characterizes total non-negativity of M e → a when a is non-decreasing. Here we fully extend this result. For arbitrary real sequences a and e , we give a condition that can be checked in O (n 2) time that determines whether M e → a is totally non-negative. When M e → a is totally non-negative, we witness this with a planar network whose weights are non-negative and whose path matrix is M e → a. When it is not, we witness this with an explicit negative minor. [ABSTRACT FROM AUTHOR]

In this paper, we consider complete homogeneous symmetric functions and provide a new formula for the number of plane partitions of n. This formula expresses the number of plane partitions of n in terms of binomial coefficients as a sum over all the partitions of n, considering the multiplicity of the parts greater than one. We obtain similar results for the number of strict plane partition of n and the number of symmetric plane partitions of n. [ABSTRACT FROM AUTHOR]