All Questions
11 questions
4
votes
2
answers
279
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Combinatorial representation of function
Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0....
- 83
4
votes
0
answers
139
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A combinatorial proof of an identity of partitions (Macdonald I.5)
This is a statement from Symmetric Functions and Hall Polynomials by Macdonald:
$\sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2$ where $\lambda$ denotes a partition or a Young diagram, and for each ...
- 820
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
2
votes
1
answer
314
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Finding Littlewood-Richardson coefficients without using identities
The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq |Q| + |P|...
6
votes
1
answer
392
views
hook-length formula: "Fibonaccized": Part II
This is a natural follow-up to my previous MO question, which I share with Brian Hopkins.
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \...
- 43.2k
15
votes
2
answers
1k
views
hook-length formula: "Fibonaccized" Part I
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \lambda$, define the hook numbers $h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1$ where $\...
- 43.2k
7
votes
1
answer
832
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Identity involving a sum over all partitions of $n$
In some work I've been doing on the cohomology of the moduli space of curves, the following identity has come up:
$$\prod_{i=1}^n \frac{x^{i-1}}{x^i-1} = \sum_{(a_1^{r_1},\ldots,a_{\ell}^{r_{\ell}}) \...
- 44.8k
28
votes
3
answers
1k
views
Inequality for hook numbers in Young diagrams
Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$, define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
- 17k
2
votes
0
answers
89
views
Rank-unimodality and Sperner property of higher dimensional partitions
I did a google-search but have not been able to find much reference on this problem. So I am asking it here hoping to get some information.
Consider the set of all 4-dimensional Ferrer's diagram ...
- 1,305
8
votes
1
answer
2k
views
A remarkable sum over partitions
While studying some seemingly unrelated topological questions, I have experimentally discovered what appears (to me) to be a remarkable sum over partitions. I was wondering if anyone knows how to ...
- 83
8
votes
3
answers
2k
views
Computing the lexicographic indices of integer partition
If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that ...
- 245