A probabilistic interpretation of the Gaussian binomial coefficients

2017 ◽  
Vol 54 (4) ◽  
pp. 1295-1298 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Linglong Yuan
Keyword(s):  
Random Walk ◽  
Simple Proof ◽  
Lattice Point ◽  
Binomial Coefficients ◽  
Probabilistic Interpretation ◽  
Gaussian Binomial Coefficients

Abstract We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.

scholarly journals Gaussian Binomial Coefficients with Negative Arguments

2019 ◽  
Vol 23 (3-4) ◽  
pp. 725-748
Author(s):  
Sam Formichella ◽  
Armin Straub
Keyword(s):  
Binomial Coefficients ◽  
Gaussian Binomial Coefficients

On a Result of Aleliunas et al. Concerning Random Walks on Graphs

1990 ◽  
Vol 4 (4) ◽  
pp. 489-492 ◽  
Author(s):  
José Luis Palacios
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Proof ◽  
Hitting Times ◽  
Initial State ◽  
Random Walks On Graphs ◽  
Minimum Number

Aleliunas et al. [3] proved that for a random walk on a connected raph G = (V, E) on N vertices, the expected minimum number of steps to visit all vertices is bounded by 2|E|(N - 1), regardless of the initial state. We give here a simple proof of that result through an equality involving hitting times of vertices that can be extended to an inequality for hitting times of edges, thus obtaining a bound for the expected minimum number of steps to visit all edges exactly once in each direction.

scholarly journals -DEFORMED RATIONALS AND -CONTINUED FRACTIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
SOPHIE MORIER-GENOUD ◽  
VALENTIN OVSIENKO
Keyword(s):  
Continued Fractions ◽  
Jones Polynomial ◽  
Total Positivity ◽  
Recursive Formula ◽  
Rational Numbers ◽  
Binomial Coefficients ◽  
Indecomposable Representation ◽  
Pascal Triangle ◽  
Rational Knots ◽  
Gaussian Binomial Coefficients

We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

scholarly journals Some identities of Gaussian binomial coefficients

2022 ◽  
Vol Accepted manuscript ◽  
Author(s):  
Tian-Xiao He ◽  
Anthony G. Shannon ◽  
Peter J.-S. Shiue
Keyword(s):  
Binomial Coefficients ◽  
Recursive Sequences ◽  
Fibonomial Coefficients ◽  
Complete Functions ◽  
Gaussian Binomial Coefficients

In this paper, we present some identities of Gaussian binomial coefficients with respect to recursive sequences, Fibonomial coefficients, and complete functions by use of their relationships.

Gorenstein cohomology of N-complexes

2019 ◽  
Vol 19 (09) ◽  
pp. 2050174
Author(s):  
Bo Lu ◽  
Zhenxing Di
Keyword(s):  
Projective Dimension ◽  
Complex Case ◽  
Binomial Coefficients ◽  
Cohomology Groups ◽  
Injective Dimension ◽  
Relative Cohomology ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective ◽  
Gaussian Binomial Coefficients

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.

scholarly journals A Simple Proof for a Forbidden Subposet Problem

10.37236/7680 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ryan R. Martin ◽  
Abhishek Methuku ◽  
Andrew Uzzell ◽  
Shanise Walker
Keyword(s):  
Simple Proof ◽  
Simple Approach ◽  
The Other ◽  
Binomial Coefficients ◽  
Other Hand

The poset $Y_{k, 2}$ consists of $k+2$ distinct elements  $x_1$, $x_2$, \dots, $x_{k}$, $y_1$, $y_2$, such that $x_1 \le x_2 \le \cdots \le x_{k} \le y_1$, $y_2$. The poset $Y'_{k, 2}$ is the dual poset of $Y_{k, 2}$. The sum of the $k$ largest binomial coefficients of order $n$ is denoted by $\Sigma(n,k)$. Let $\mathrm{La}^{\sharp}(n,\{Y_{k, 2}, Y'_{k, 2}\})$ be the size of the largest family $\mathcal{F} \subset 2^{[n]}$ that contains neither $Y_{k,2}$ nor $Y'_{k,2}$ as an induced subposet. Methuku and Tompkins proved that $\mathrm{La}^{\sharp}(n, \{Y_{2,2}, Y'_{2,2}\}) = \Sigma(n,2)$ for $n \ge 3$ and conjectured the generalization that if $k \ge 2$ is an integer and $n \ge k+1$, then $\mathrm{La}^{\sharp}(n, \{Y_{k,2}, Y'_{k,2}\}) = \Sigma(n,k)$. On the other hand, it is known that $\mathrm{La}^{\sharp}(n, Y_{k,2})$ and $\mathrm{La}^{\sharp}(n, Y'_{k,2})$ are both strictly greater than $\Sigma(n,k)$. In this paper, we introduce a simple approach, motivated by discharging, to prove this conjecture.  

Sign in / Sign up

Export Citation Format

Share Document