Related papers: Irrational proofs for three theorems of Stanley
Stanley defined a partition function t(n) as the number of partitions $\lambda$ of n such that the number of odd parts of $\lambda$ is congruent to the number of odd parts of the conjugate partition $\lambda'$ modulo 4. We show that t(n)…
Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if…
In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and…
It is known that the Ehrhart polynomials of cross-polytopes, as well as of pyramids over them, have positive coefficients. We give a combinatorial proof of this fact by showing that a scaled version of the Ehrhart polynomials are generating…
Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are…
In Ehrhart theory, the well-known sign pattern problem asks: given a positive integer $d\geq 3$ and integers $1 \leq i_1 < \cdots < i_k \leq d-2$, does there exist a $d$-dimensional integral polytope $\mathcal{P}$ such that in its Ehrhart…
In a previous paper, we showed how to use the Ehrhart function $L_P(s)$, defined by $L_P(s) = \#(sP \cap \mathbb Z^d)$, to reconstruct a polytope $P$. More specifically, we showed that, for rational polytopes $P$ and $Q$, if $L_{P + w}(s) =…
In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all…
The exponential generating functions of {n^(n+m)} for arbitrary integer m are expressed as rational functions of the e.g.f. of {n^(n-1)} [the tree function] and then of the e.g.f. of {n^n} [the endofunction function]. The coefficients in…
An important question arising from the Frobenius Coin Problem is to decide whether or not a given monetary sum S can be obtained from N coin denominations. We develop a new Generating Function G(x), where the coefficient of x^i is equal to…
Plethysm coefficients $\mathsf{a}_{\mu[\nu]}^\lambda$ are the structure coefficients of the plethysm of Schur functions $s_\mu[s_\nu] = \sum_{\lambda} \mathsf{a}_{\mu[\nu]}^\lambda s_\lambda$. We study a bivariate generating function of…
We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind $ \JS(n+k,n;z)$ by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting…
We show that the Stanley's Conjecture holds for an intersection of three monomial primary ideals of a polynomial algebra S over a field.
Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a rational function with integer coefficients…
We study the Ehrhart theory of quadratic irrational polytopes that undergo vector dilations. That is, for a given polytope with vertices in $\mathbb{Q}(\sqrt{D})$, and a different dilation factor for each facet, we show that the leading…
We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary…
We complete the proof of the fact that all principal permutation classes generated by a pattern longer than two have a nonrational generating function.
In this paper we extend the Smarandache function from the set $N*$ of positive integers to the set $Q$ pf rational numbers. Using the inverse formula, this function is also regarded as a generating function. We put in evidence a procedure…
We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.
We introduce the theory of $(\mathcal{P},\rho)$-partitions, depending on a poset $\mathcal{P}$ and a map $\rho$ from $\mathcal{P}$ to positive integers. The generating function $\mathfrak{F}_{\mathcal{P},\rho}$ of…