Related papers: Modular Fuss-Catalan numbers
Planar commutative n-complex numbers of the form u=x_0+h_1x_1+h_2x_2+...+h_{n-1}x_{n-1} are introduced in an even number n of dimensions, the variables x_0,...,x_{n-1} being real numbers. The planar n-complex numbers can be described by the…
We introduce a new family $\mathcal{A}_{n,k}$ of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for $k=1$, this family is equal to a multivariate…
A coding method using binary sequences is presented for different computation problems related to Catalan numbers. This method proves in a very easy way the equivalence of these problems.
I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal…
This paper uses monads and comonads to establish a certain type of equivalence between two subcategories, one reflective and one coreflective, in a category whose objects represent compactifications of non-compact locally compact Hausdorff…
The connection between the commutativity of a family of $n\times n$ matrices and the generalized joint numerical ranges is studied. For instance, it is shown that ${\cal F}$ is a family of mutually commuting normal matrices if and only if…
Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and…
We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary $C^*$-algebra (respectively, von Neumann algebra), and provide applications of these results to the…
A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in…
Clones of functions play a foundational role in both universal algebra and theoretical computer science. In this work, we introduce clone merge monoids (cm-monoids), a unifying one-sorted algebraic framework that integrates abstract clones,…
We study and classify faithfully balanced modules for the algebra of lower triangular $n$ by $n$ matrices. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers.…
Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $\delta$ corresponds to the…
Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this…
We present the new combinatorial class of product-coproduct prographs which are planar assemblies of two types of operators: products having two inputs and a single output and coproducts having a single input and two outputs. We show that…
We present determinantal representations of the Catalan numbers, k-Fuss-Catalan numbers, and its generalized number. The entries of the normalized Hessenberg matrices are the binomial coefficients that related with the enumeration of…
We construct a family of $S_n$ modules indexed by $c\in\{1,\dots,n\}$ with the property that upon restriction to $S_{n-1}$ they recover the classical parking function representation of Haiman. The construction of these modules relies on an…
Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…
A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\displaystyle{\frac{n}{c}}$. For any integer $k$, the function $\sigma_k^*$ is a multiplicative arithmetic function defined so that…
We encode the real homotopy type of an $n$-dimensional $(r-1)$-connected compact manifold $M$, $ r\ge 2$ into a minimal unital $C_\infty$-structure on $H^* (M,\mathbb R)$, obtained via homotopy transfer of the unital DGCA structure of the…
Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain…