Questions tagged [enumerative-combinatorics]
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539 questions
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Finding a sum that is always divisible by $1+2+3+\cdots+n$ [closed]
There are $n$ consecutive integers $m,m+1, \ldots, m+n-1$. Prove that you can choose some nonempty subset of these numbers whose sum is divisible by $1+2+\dots+n$.
- 77
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3
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A question on the plethysm of complete symmetric functions
Based on some small calculations in SageMath, I conjectured that the Schur expansion of $h_n[h_k]$ contains $s_{(k,k,...,k)}$ if and only if $k$ is even. For example, this is easily seen when $n=2$. ...
- 419
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3
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Peculiar exception in the number of distinct values taken by the sums of the 6th degree roots of unity
For a nonnegative integer $n$, let $N_n$ be the number of distinct values taken by the sums of $n$ 6th-degree roots of unity (with repetitions). First few counts are $N_0=1$, $N_1=6$, $N_2=19$, and ...
- 38.4k
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Algorithms to count restricted injections
Let the following data be given.
Two positive integers $m$ and $n$.
A family of sets $B_a \subseteq \{1, \dots, n\}$ (for $a \in \{1, \dots, m\}$).
The task is to count the number $N$ of injective ...
- 143
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Counterexample to a generalization of Frankl's union-closed sets conjecture
We call a family of sets $\mathcal{F}$ is weakly union-closed if for all $A,B\in\mathcal{F}$ such that $A\cap B=\varnothing$, we have $A\cup B\in\mathcal{F}$.
Conjecture: For finite weakly union-...
- 1,558
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Is there a canonical name for this variant of the rook polynomial?
Let $\mathcal{P}$ be a polyomino. Two or more rooks on $\mathcal{P}$ are called non-attacking if no path of edge-adjacent cells of $\mathcal{P}$ connects any pair of them along a row or a column.
Let $...
- 1,357
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$\mathcal{P}$-rook polynomial of a grid
The $\mathcal{P}$-rook polynomial of a polyomino $\mathcal{P}$ is
$$
r_\mathcal{P}(T) = \sum_{k=0}^{r(\mathcal{P})} r_k(\mathcal{P})\ T^k,
$$ where $r_k(\mathcal{P})$ is the number of ways to place $...
- 1,357
2
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1
answer
360
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An approach to a generalization of Frankl's union-closed sets conjecture
Let $I$ be a non-empty finite set, $\mathcal{F}$ be a non-empty union-closed family of subsets of $I$ except the empty set and $n$ real numbers $x_i\geq1,i\in I$. Let $d_i=\sum_{i\in J,J\in\mathcal{F}}...
- 1,558
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Counting edge-simple walks of length $k$ in the complete graph $K_n$ that cover the whole graph
Let $K_n$ be the complete graph on $n$ vertices. I am interested in counting the number of walks of length $k$ in $K_n$ with the following constraint:
Edges may not be repeated (each edge is used at ...
- 685
1
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1
answer
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Counting multidimensional arrays up to reindexing and relabeling
Define a $d_0\times d_1\times\cdots\times d_{k-1}$-grid to be a surjective function $G$ with domain $\prod_{i=0}^{k-1}\{0,1,\dots,d_i-1\}$. A $d_0\times d_1\times\cdots\times d_{k-1}$-grid can also be ...
- 11
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1
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Using Pascal's identity and summation of binomial coefficients: proof of a weighted sum formula
I am trying to prove the following identity involving binomial coefficients.
It is known that:
-For any $\ell \in \mathbb{N}$ and any $r \geq \ell$, we have the summation formula:
$$
\sum_{j=\ell}^r\...
- 257
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Is the rook polynomial of a convex polyomino determined?
Consider a convex polyomino, which can be uniquely defined by its horizontal and vertical projection vectors.
More precisely, the horizontal projection vector is
$h = (h_1, h_2, \dots, h_m)$,
where $...
- 1,357
1
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382
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More about the algebraic strengthening of Frankl's union-closed conjecture
Continue my previous question, consider the first conjecture:
Let $\mathcal{F}$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: ...
- 1,558
5
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Number of compact connected Lie groups of given dimension
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\HSpin{HSpin}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{...
- 409
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212
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Closed form expression for the enumeration of permutations with no fixed points and only nontrivial intervals
Let $A_n$ be the number of permutations $\pi$ of $[n]=\{1,2,\ldots,n\}$ such that $\pi(i)\neq i$ for all $i \in [n]$ and $|\pi(i+1) - \pi(i)| > 1$ for all $i \in [n-1]$. So $A_n$ counts the ...
- 557
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111
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Number of central $3$-dim hyperplane arrangements in generic position
Let $a(n)$ be the number of equivalence classes of $3$-dimensional central arrangements of $n$ hyperplanes in general position.
I believe that $a(n)= 1$ for $n \leq 5$ and $a(6)=3$. Is this correct? ...
- 1,882
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370
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Descent sets and inverse descent sets of a permutation
Let $w=a_1 a_2\cdots a_n\in S_n$, the symmetric group of all
permutations of $1,2,\dots,n$. The descent set $D(w)$ is defined by
$D(w)=\{1\leq i\leq n-1\,:\, a_i>a_{i+1}\}$. Let $f(n)$ be the ...
- 53.6k
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Scaling in unbalanced gale–shapley proposals
Let $G = (A, B, E)$ be a bipartite graph representing a matching market with $|A| = n+1$ and $|B| = n$. Each vertex $a \in A$ has a preference list that is a uniformly random permutation of $B$, and ...
user551082
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Specific counting function in $\mathbb{Z}_2^n$
Let $A, B, C \subseteq \mathbb{Z}_2^n$. Define $r(A, B, C) := |\{(a, b, c) \in A \times B \times C : a + b = c\}|$.
For $1 \leq a \leq 2^n - 1$, let $A_a \subseteq \mathbb{Z}_2^n$ denote the set ...
- 448
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148
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Formula for sum involving products of (symplectic) Schur functions
This question is a continuation of a question asked yesterday which had a very nice answer.
Consider the summation
$$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
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155
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A question about decomposing root system $A_{n}$
Denote $\Phi(n)$ as the root system of Lie algebra $\mathfrak{g}$ of type $A_{n}$. Call a disjoint union $\Phi(n) = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if each $\Phi_{k}$ is ...
- 635
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0
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196
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LIS-based permutation property
Let $S_n$ be the set of all permutations of $\{1, \ldots, n\}$, thereafter treated as integer sequences. Let $A_n$ be the set of all such permutations $\sigma \in S_n$ that we can choose two ...
- 5,752
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Number of points covered by $2n$ hyperplanes in $\mathbf{F}_p^n$
For a prime $p$, fix two bases $U=\{v_1,\dots,v_n\}$ and $W=\{w_1,\dots,w_n\}$ of the vector space $V=\mathbf{F}_p^n$. We may assume $U$ is the standard basis without loss of generality.
For $s_1,\...
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On the parity of the sum $\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$
QUESTION. Let $p$ be an odd prime and let $a,b,c\in\mathbb Z$. How to determine the parity of the sum
$$S_p(a,b,c)=\sum_{1\le j<k\le p-1\atop p\nmid aj^2+bjk+ck^2}(aj^2+bjk+ck^2)$$
in terms of $a,b,...
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Enumerative number theory term searching [closed]
Given $a,b$ positive numbers such that $gcd(a,b)=1$.Prove that there are infinitely many $n$ positive integers such that $x_n=a+nb$ sequence has many terms such that it is not divisible by any prime'...
