Questions tagged [enumerative-combinatorics]
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504 questions
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Non-enumerative proof that there are many derangements?
Recall that a derangement is a permutation $\pi: \{1,\ldots,n\} \to \{1,\ldots,n\}$ with no fixed points: $\pi(j) \neq j$ for all $j$. A classical application of the inclusion-exclusion principle ...
- 114k
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votes
3
answers
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Distinct numbers in multiplication table
Consider the multiplication table for the numbers $1,2,\dots, n$. How many different numbers are there? That is, how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there?
I'm ...
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votes
5
answers
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How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
- 151k
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votes
6
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Combinatorial Morse functions and random permutations
This question has its origin in combinatorial topology. In the 90s R. Forman proposed a discrete counterpart of Morse theory. In his case, a Morse function on a triangulated space is a function ...
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votes
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Number of collinear ways to fill a grid
A way to fill a finite grid (one box after the other) is called collinear if every newly filled box (the first excepted) is vertically or horizontally collinear with a previously filled box. See the ...
25
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3
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The verbs in combinatorics: Enumerating, counting, listing and all that
Two closely related, but different tasks in combinatorics are
determining the number of elements in some set $A$, and
presenting all its elements one by one.
Question: What are some works in ...
- 4,219
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3
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What can be proved about the Ramanujan conjecture using elementary means?
The Ramanujan conjecture states that the coefficients $\tau(n)$ in the identity
$$q\prod_{m=1}^\infty(1-q^m)^{24}=\sum_{n=1}^\infty\tau(n)q^n$$
satisfy the inequality $|\tau(n)|\leq d(n)n^{11/2}$, ...
- 29k
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3
answers
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Proofs of parity results via the Handshaking lemma
I particularly like the following strategy to prove that the number of some combinatorial objects is even: to construct a graph, in which they correspond to vertices of odd degree (=valency).
Let me ...
23
votes
2
answers
1k
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Fibonacci, compositions, history
There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets):
a) compositions with parts from {1,2}
(e.g., 2+2 = 2+1+1 = ...
- 4,589
21
votes
1
answer
1k
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A strange sum over bipartite graphs
While mucking around with some generating functions related to enumeration of regular bipartite graphs, I stumbled across the following cutie. I wonder if anyone has seen it before, and/or if anyone ...
- 37.7k
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1
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Combinatorial proof of a certain binomial identity
Let $n$, $p$, $q$ be non-negative integers. Then
$$
\sum_{k=0}^n{2k+2p\choose k+p,k,p}{2(n-k)+2q\choose n-k+q,n-k,q}=4^n{2p\choose p}{2q\choose q}{n+p+q\choose n}.\tag{$\heartsuit$}\label{heart}
$$
In ...
- 109k
21
votes
1
answer
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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 ...
- 3,209
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4
answers
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Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
It is a nice exercise with rational generating functions (or equivalently, ...
- 24.2k
20
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2
answers
920
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Counting problems where unlabeled is easier than labeled
I was encouraged to post this question by Jim Propp during a meeting of the Cambridge Combinatorics and Coffee Club. It is a counterpoint to the MathOverflow question "Counting Problems where Labeled ...
- 24.2k
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3
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A generalization of the triangle counting problem for simple weighted graphs
One nice identity is that $$\operatorname{tr}(A^3)/6$$ counts the number of triangles of a graph with adjacency matrix $A$. It also implies that triangle counting in a graph can be performed in sub-...
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3
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Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
- 32.1k
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A combinatorial interpretation for $n$-ary trees for negative $n$
The ordinary generating function $T_n=T_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T_n=1+xT_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be ...
- 1,190
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2
answers
991
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A conjecture harmonic numbers
I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven).
From the Online Encyclopedia of Positive Integers we have:
$a(n)$ ...
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1
answer
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Salié permutations and fair permutations
In October 2010, I published a Monthly problem that introduced the concept of a fair permutation, which is a permutation $\pi$ such that for every $i$, either $\pi(i) > i$ and $\pi^{-1}(i) > i$, ...
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2
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Is this similar to a known combinatorial identity?
(Apologies if this is too obscure.)
In joint work with Izzet Coskun we came across the following kind of combinatorial identity, but we weren't able to prove it, or to identify what kind of identity ...
user5117
18
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0
answers
987
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"Special" meanders
One of the open problems in combinatorics is enumeration of meanders.
Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand.
Since my ...
- 18.4k
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1
answer
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Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 128k
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2
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How many triangulations of the genus $g$ surface on $n$ vertices?
By "a triangulation of $X$", I mean a simplicial complex whose geometric realization is homeomorphic to $X$. Tutte showed that the number of combinatorially distinct triangulations $t(n)$ of the $2$-...
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2
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Is there any meaning to a "nice bijective proof?"
From Zeilberger's PCM article on enumerative combinatorics:
The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal ...
- 12.6k
16
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1
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Existence of a faithful irreducible representation using Möbius function
Let $G$ be a finite group, $L(G)$ its subgroup lattice and $\mu$ the Möbius function.
Consider the Euler totient of $G$ defined as follows:
$$ \varphi(G) = \sum_{H \le G}\mu(H,G) |H| $$
Let $X=\{M_1, \...
15
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4
answers
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Collecting alternative proofs for the oddity of Catalan
Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
- 43.1k
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A rather curious identity on sums over triple binomial terms
While exploring the Baxter sequences from my earlier MO post, I obtained a rather curious identity (not listed on OEIS either). I usually try to employ the Wilf-Zeilberger (WZ) algorithm to justify ...
- 43.1k
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2
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Convergency radius of the generating series for A93637
Sequence A93637 of the OEIS (https://oeis.org/A093637) starting as $1,1,2,4,9,20,49,117,297,746,1947,\ldots$ is defined by
the coefficients $a_0,a_1,\ldots$ of the unique formal power series
defined ...
- 17.9k
15
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2
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910
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Sequences that don't count algebraic structures on finite sets
People count $n$-element groups, $n$-element monoids, $n$-element commutative monoids, etcetera - always up to isomorphism. The algebraic structures I've listed, and many more, are studied ...
- 22.2k
15
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2
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Generating functions for objects with irrational sizes
A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able ...
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1
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Character theoretic proof of the Littlewood–Richardson rule?
The Littlewood–Richardson coefficients are the multiplicities
$$
c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu))
$$
and the Littlewood–Richardson rule says that ...
- 1,413
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5
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Are there more connected or disconnected graphs on $n$ vertices?
Suppose we are talking about graphs with $n$ labeled vertices. Which graphs are more common: connected or disconnected?
- 1,765
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3
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On the finite sum of reciprocal Fibonacci sequences
I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{$*$}$$ where $\lfloor x \rfloor$ is th floor function.
The Fibonacci ...
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Are the asymptotics of A003238 known?
Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is ...
- 1,305
14
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2
answers
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Number of matchings of even cycles
By doing some calculations on the generating function of matching polynomials of cycles I made the following interesting observation:
For all positive integers $n>1$ and $k <n $, the number of ...
- 4,484
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1
answer
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Bijective proof of recurrence for rooted unlabeled trees
Would've been a better question for Christmas than Thanksgiving, but alas...
Let $t_n$ denote the number of rooted, unlabeled trees on $n$ vertices (OEIS A000081). These are the isomorphism classes of ...
- 24.2k
14
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0
answers
270
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A symmetry of lattice paths
The number of $n$-step NSEW lattice paths from $(0,0)$ to $(a,b)$ that intersect the line $y=k$ precisely $t$ times is independent of $k$, for $0\leq k\leq b$, where we assume $b\geq0$ for simplicity.
...
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Is the following invariant of rooted trees a complete invariant?
Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new vertex ...
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2
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an identity for a sum over partitions
Write an integer partition $\lambda\vdash n$ in two different ways:
(1) $\lambda=\lambda_1\geq\lambda_2\geq\lambda_3\cdots\geq\lambda_k\geq1$
(2) $\lambda=1^{m_1}2^{m_2}3^{m_3}\cdots n^{m_n}$ for ...
- 43.1k
13
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2
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Two interpretations of a sequence: an opportunity for combinatorics
The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function
$$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look ...
- 43.1k
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1
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The Möbius number of the nonabelian finite simple groups
Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(...
13
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1
answer
728
views
Maximum number of distinct diagonals generated by permutations
Given a matrix $A \in \{0,1\}^{n \times n}$, let $diag(A)$ be the set of vectors $D \in \{0,1\}^n$ that are the diagonal of one of the $n!$ matrices obtained from $A$ via row permutations.
What is ...
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1
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Erdős multiplication problem revisited
This is a well-known problem and is about counting the number of distinct numbers in the $n \times n$ multiplication table.
The very problem has been discussed in-depth and, as such, I require no ...
user56218
13
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1
answer
564
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Coincidences between average Catalan tableaux
There are Catalan number $C_n$ of standard Young tableaux of shape $(n,n)$, which we view as $2\times n$ matrices. Denote by $P_n$ the average of these matrices:
$$
P_n \, := \, \frac{1}{C_n} \, \...
- 17k
13
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1
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Two to the power of a triangular number: bijections
The numbers $2^{n(n+1)/2}$ come up in various enumerative contexts. In addition to the trivial example (bit-strings of length $n(n+1)/2$) and the old example of domino tilings of Aztec diamonds (...
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Succesful applications of algebra in combinatorics
Hi. This may be a very general question.
Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?
If yes, could somebody ...
13
votes
0
answers
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Reference request: exponential growth rates of subword-closed languages are integers
For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $n$. The word $u$ is a subword of $w$ if $u$ can be obtained from $w$ by deleting letters (...
- 2,339
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3
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A "quantum" identity: in search of a proof -Part II
As usual, denote $[n]_q=1+q+\cdots+q^{n-1}=\frac{\,\,1-q^n}{1-q}$ and $[n]_q!=[1]_q[2]_q\cdots[n]_q$. Furthermore, we write
$$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\cdot[n-k]_q!}.$$
As a follow up on this ...
- 43.1k
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An interesting identity: in search of a proof -Part I
I like the following binomial identity in that the RHS extracts the indeterminate $w$ from the LHS.
Question. Can you show that
$$\sum_{k=0}^n\binom{x+kw}k\binom{y-kw}{n-k}=\sum_{k=0}^n\binom{x+y-...
- 43.1k
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votes
5
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Number of permutations with a specified number of fixed points
Let $F(k,n)$ be the number of permutations of an n-element set that fix exactly $k$ elements.
We know:
$F(n,n) = 1$
$F(n-1,n) = 0$
$F(n-2,n) = \binom {n} {2}$
...
$F(0,n) = n! \cdot \sum_{k=0}^n \...
- 9,720