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Results tagged with co.combinatorics
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user 81507
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
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Intersection property of Szemerédi's regularity condition
We adopt common notations in the study of Szemerédi's regularity lemma and only focus on simple graph $G(V,E)$. For any two disjoint vertex sets $A,B\subset V$, we say the pair $(A,B)$ is $\varepsilon …
- 139
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The restriction of any Freiman homomorphism of order 2 to $\{kd:-l\leq k \leq l\}$ is linear?
Let $B$ be a subset of $\mathbb{Z}_N$, where $\mathbb{Z}_N$ is the group of all congruence classes mod $N$. The function $\phi:B\rightarrow \mathbb{Z}_N$ is said to be a Freiman homomorphism of order …
- 139
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How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lam...
Let $\lambda$ be a partition of an positive integer $n$, it can be presented as $\lambda=(\lambda_{1},\lambda_2,\cdots,\lambda_l)$ such that $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_l>0$, or $\lam …
- 139
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286
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How to prove that $\sum_{i=0}^n\frac{(a;q)_i}{(q;q)_i}\frac{(b;q)_{n-i}}{(q;q)_{n-i}}a^{n-i}...
By Cauchy identity, $${}_1\phi_0(a;—;q,z)=\sum_{n\geq0}\frac{(a;q)_n}{(q;q)_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_\infty},\quad|z|<1,|q|<1,$$
we can obtain the $q-$analogue of $(1-z)^{-a}(1-z)^{-b}=(1-z) …
- 139
1
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Inequality about the minimum vertex degree in $k$-uniform hypergraphs
Let $H=(V,E)$ be a $k$-uniform hypergraph with $n$ vertices, that is, $V:=V(H)$ is a $n$-element finite set of vertices and $E:=E(H)\subset\binom{V}{k}$ is a family of $k$-element subsets of $V$.
Gi …
- 139