All Questions
9 questions
2
votes
3
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395
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Question for averaging the overall quantities by averaging a part
There is a question: If integers $a$ and $b$ satisfy the following properties: for any $a$ real numbers, we can do an operation to average $b$ of them to the same quantities, and we can do a finite ...
- 843
0
votes
0
answers
80
views
Minimizing coefficients in a product related to the Rogers Ramanujan identity
Start with the product for partitions into parts congruent to $1$ or $4$ modulo $5$:
$(1 + x + x^2 + x^3 + ...)(1 + x^4 + x^8 + x^{12} +...)(1 + x^6 + x^{12} + x^{18} +...)$...
Now replace some of the ...
- 211
6
votes
1
answer
392
views
hook-length formula: "Fibonaccized": Part II
This is a natural follow-up to my previous MO question, which I share with Brian Hopkins.
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \...
- 43.2k
15
votes
2
answers
1k
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hook-length formula: "Fibonaccized" Part I
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \lambda$, define the hook numbers $h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1$ where $\...
- 43.2k
5
votes
1
answer
610
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Eulerian ordering of the integers modulo n
Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$.
An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that:
$$\forall i \le n \ \forall j&...
5
votes
1
answer
261
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Divisibility labeling on a boolean lattice and nonzero Euler totient
Let $B_n$ be the subset lattice of $\{1,2, \dots , n \}$, also called the boolean lattice of rank $n$.
A labeling $f: B_n \to \mathbb{N}_{\ge 1}$ is called acceptable if $\forall a,b \in B_n$:
...
15
votes
1
answer
417
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What is the smallest cardinality of a self-linked set in a finite cyclic group?
A subset $A$ of a group $G$ is defined to be self-linked if $A\cap gA\ne\emptyset$ for all $g\in G$. This happens if and only if $AA^{-1}=G$.
For a finite group $G$ denote by $sl(G)$ the smallest ...
- 41.8k
7
votes
1
answer
832
views
Identity involving a sum over all partitions of $n$
In some work I've been doing on the cohomology of the moduli space of curves, the following identity has come up:
$$\prod_{i=1}^n \frac{x^{i-1}}{x^i-1} = \sum_{(a_1^{r_1},\ldots,a_{\ell}^{r_{\ell}}) \...
- 44.8k
16
votes
0
answers
454
views
A Product Related to Unrestricted Partitions
Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
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