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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2
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43 views

Permutation equivalence classes with kendall-tau distance

I asked this question on Stack Exchange a few days ago with no help. I am not sure if the question seems too trivial or of not enough general interest to get any attention. Anyway, any sort of ...
4
votes
1answer
106 views

Number of bases of a matroid

I would like to know the minimum number of bases of a matroid of rank $k$ and $n$ elements, knowing that each singleton is independent. At least for small ranks.
3
votes
0answers
81 views

How many different sums of parts of a vector

I hope this question isn't too basic for MO. I also asked it on math.se previously. This mathematical puzzle was given to me by a friend a while ago and I can't work out how to solve it. Does anyone ...
0
votes
0answers
58 views

Showing that a sequence of random variables with increasing expected value converges to a Poisson random variable

Consider a sequence $\{X_n\}_{n \geq 1}$ of nonnegative, integer-valued random variables. For any random variable $Y$ and $k \geq 1$, let $(Y)_k = Y(Y-1)(Y-2)\dots(Y-k+1)$ be the $k^\mathrm{th}$ ...
14
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0answers
176 views

An algebraic strengthening of the Saturation Conjecture

The Saturation Conjecture (proved by Knutson-Tao) asserts that $c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq 0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ ...
14
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0answers
117 views

Minimal number of intersections in a convex $n$-gon?

For a convex polygon $P$, draw all the diagonals of $P$ and consider the intersection points made by those diagonals. Let $f(n)$ be the minimal number of such intersections where $P$ ranges over all ...
2
votes
1answer
111 views

Is the top interval of a finite distributive lattice an hypercube lattice?

Let $(L,\wedge,\vee)$ be a finite distributive lattice. Let $M$ be the (unique) maximum element. An element $a \in L$ is called maximal if $a \le a' < M$ implies $a = a'$. Let $b = ...
1
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0answers
20 views

What is the minimum number of filled cells in a partial Latin rectangle with autotopism group $\cong C_2$ and autoparatopism group $\cong S_3$?

Definitions: a partial Latin rectangle is an $r \times s$ matrix containing symbols from $[n] \cup \{\cdot\}$ such that each row and each column contains at most one copy of any symbol in $[n]$. The ...
2
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0answers
51 views

A question on the behavior of intersections of certain block design

Let $[d]$ be a universe and $S_1, \dots, S_m$ be an $(\ell, a)$-design over $[d]$ which means that: $\forall i \in [m], S_i \subseteq [d], |S_i|=\ell$. $\forall i \neq j \in [m]$, $|S_i \cap S_j| ...
0
votes
0answers
61 views

A constrained minimum edge coloring

Is minimum number of colors needed to color edges of complete graph $K_n$ so that every even simple cycle contains at least one color assigned to odd number of edges at most $\beta n$ where ...
6
votes
1answer
126 views

Writing the quotient of solutions of linear ODEs as the solution of a nonlinear ODE

Let $A(x)$ and $B(x)$ be solutions to homogeneous linear ODEs with polynomial coefficients, i.e., $A(x)$ satisfies $$p_KA^{(K)} + p_{K-1}A^{(K-1)} + \cdots + p_1A' + p_0A=0$$ and $B(x)$ satisfies ...
1
vote
1answer
290 views

Analytic Number Theory without Pigeonhole Principle [closed]

I don't know if this is an appropriate question for this website, but I will try my luck. I am an undergraduate student, and recently I became interested in analytic number theory. When I started ...
1
vote
0answers
70 views

Efficiently counting all paths of length n in a graph with vertex visitation contraints [closed]

I have a graph G with two classes of vertices. The first class represents no resource limitation entities and can be visited an unlimited number of times in any path traversal. The second class of ...
1
vote
0answers
169 views

system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [on hold]

What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$? Define $F$ in the domain where a continuous function exists that behaves so for ...
2
votes
1answer
97 views

Inducing representations from the stabilizer of a partition

For each positive integer $i$, let $A_i$ be a fixed representation of the symmetric group $S_i$. I won't tell you exactly what $A_i$ is, but let's say that I have a very explicit description of its ...
1
vote
0answers
216 views

Incidence geometry and matrices

Supposing I have a $0/1$ or $\pm1$ matrix $A$ of size $m\times n$, is there a minimum $d$ (that works for every $m\times n$ $A$) such that there exists $m$ lines $r_1,\dots,r_m$, $n$ lines ...
1
vote
1answer
304 views

Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected. The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
7
votes
1answer
454 views
+50

Counting subspaces

We are given the finite vector space $V = V(n,p) = \mathbb{F}_p^n$ and two fixed subspaces $W_1, W_2 \subseteq V$ of dimensions $m_1$, $m_2$ respectively. Suppose that the dimension of $W_1 \cap W_2$ ...
8
votes
2answers
222 views

Dividing the edges and diagonals of a polygon among disjoint sub-polygons

Let $P$ be a convex $n$-gon ($n$ is odd and $n \geq 5$). Determine the smallest $m$ such that all edges and diagonals of $P$ can be covered by the edges of $m$ convex sub-polygons of $P$ which ...
12
votes
2answers
400 views

Number of nonzero terms in polynomial expansion (lower bounds)

Let $f(x) = a_1x^{z_1} + a_2x^{z_2} + \cdots + a_kx^{z_k}$ be a polynomial with coefficients $(a_1, \ldots, a_k) \in \mathbb{F}_q^*$ and $z_i$ are distinct positive integers. If I need to compute the ...
1
vote
0answers
145 views

Nimbers and Surreal Numbers [closed]

I have been researching Combinatorial Game Theory. One common theme is the assignment of values to games in order to classify the game as a win for a specific player. One such way is class of surreal ...
5
votes
1answer
139 views

Modification of matching

Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and ...
2
votes
1answer
57 views

Images of interval edge coloring

I found out the definition of interval edge colorings, concept put by Kamalian in various papers but could not find a graph depicting an example. Where can I find pictures of explicit examples of ...
8
votes
1answer
195 views

Smallest strongly regular graph whose automorphism group is not vertex transitive?

I'm looking for a small strongly regular graph whose automorphism group is not vertex-transitive. This answer to a different question shows that the Chang graphs on 28 vertices are such graphs. Is ...
2
votes
1answer
80 views

Enumerator Polynomials for Linear Anytime Codes

Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...
3
votes
2answers
83 views

Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by \begin{eqnarray*} s_0 &=& a, \\ s_1 &=& b, \text{and} \\ s_{n+2} &=& s_{n+1} ...
12
votes
1answer
324 views

Generalized geometries

Let $S$ be a non-empty set. A geometry of type $n$ for $n\geq 1$ on $S$ (consisting of at least $n$ elements) is a set ${\mathfrak P}\subseteq {\mathcal P}(S)$ such that all members of $\mathfrak ...
8
votes
0answers
79 views

types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...
4
votes
1answer
169 views

Is the simplicial set $(\Delta_3/\partial \Delta_3)^{\Delta_1}$ finite?

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a ...
3
votes
1answer
134 views

Counting triple-free sequences

The Thue-Morse sequence is a triple-free element $(a_i)\in\{0,1\}^\mathbb{N}$ which can be constructed by iterating the transformations $0\to 01$ and $1\to 10$ starting from $0$. The beginning of the ...
5
votes
0answers
168 views

Paths in Pascal's triangle; or balanced $0-1$ initial segments

Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...
0
votes
1answer
101 views

Two graph structures on $\text{Hom}(G,H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...
14
votes
1answer
841 views

Prove that expression is integer

Numerical experiments suggest that $\binom{2m}{m + k}\cdot\frac{3m - 1 - 2k^2}{2m - 1}$ is integer for all $-m \le k\le +m$. It means that expression evaluation could be implemented very efficiently, ...
3
votes
1answer
203 views

Are there enough additive permutations?

I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other ...
4
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0answers
45 views

Existence of certain graph gadget related to coloring odd hole free graph

Wondering about the existence of graph gadget related to coloring (or 3-coloring) odd hole free graphs. Let $G$ be simple $k$-chromatic connected graph with two vertices $u,v$. Is it possible $G$ to ...
1
vote
1answer
72 views

Set counting problem with a cap on the intersection between the set and a fixed partition

Fix sets $T_1,\ldots T_m$ as a $k$-partition of $[m\cdot k]=\{1,2,\ldots,m\cdot k\}$, so that $|T_i|=k$ and $|T_i\cap T_\ell|=0$. 1) For any $j\le k$, how many sets $C\subset [m\cdot k]$ are there ...
1
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0answers
131 views

Complexity of reordering a matrix which consists independent sub matrices

Introduction: Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$. $A_x$ is the symmetric matrix of the graph $(G-x)$, ...
2
votes
0answers
42 views

Maximum cardinality general factor of a graph

Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...
0
votes
0answers
42 views

Enumeration of simple graphs with given degree distribution/sequence [duplicate]

Is there any exact formula for asymptotic/exact enumeration of simple graphs with given degree sequence? I just found some results about it, but the formula is hold on for some conditions, for example ...
1
vote
0answers
104 views

What are constructions for induced $C_5$-free graphs?

During a recent workshop, the question came up whether there are some constructions for graphs that are induced $C_5$-free, but they contain "everything else," so we don't want to forbid $C_5$'s, ...
0
votes
2answers
334 views

Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns. Consider $\mathscr{M}[m^\sigma]$ to be collection of ...
6
votes
1answer
101 views

Counting equivalence relations with marked classes

The number of equivalence relations on a set of $n$ elements is the Bell number $B_n$. If we wish to count the number of equivalence classes on a set of $n$ elements where one of the classes is ...
4
votes
0answers
85 views

Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim: Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...
10
votes
6answers
930 views

Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said: Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...
2
votes
2answers
211 views

binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
3
votes
1answer
84 views

$P_3$-factors for 3-regular, 3-connected cubic graphs

Suppose that $G=(V,E)$ is a simple graph. We know if $G$ is 3-regular, 3-connected and $|V|=4k$ for some $k\in \mathbb{N}$, then $G$ has a $P_4$-factor. Question. Let $G=(V,E)$ be 3-regular, ...
2
votes
2answers
176 views

Simultaneous lcms

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, ...
16
votes
0answers
291 views

Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
7
votes
2answers
267 views

When is a sequence the sum of two Beatty sequences?

In other words, given a sequence $(s_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that $$s_n = \lfloor un\rfloor + \lfloor vn\rfloor$$ for every positive integer $n$? ...
1
vote
1answer
181 views

Counting matrices of special types

How many symmetric and non-symmetric $n\times n$ matrices with $0/1$ entries are there such that every row is distinct and every column is distinct? (I am looking for a proof as well). If only every ...