All Questions
Tagged with integer-partitions algebraic-combinatorics
10
questions
-1
votes
1
answer
83
views
How many ways can someone choose a permutation $w $ and color each one of the integers [n] so that the minimum element of every cycle of w is white?
In how many ways can someone choose a permutation $w \in S_n$ and color each one of the integers $1,2,\ldots,n$ white, yellow or blue so that the minimum element of every cycle of $w$ is white?
...
user732679
1
vote
1
answer
74
views
What is the Lie theoretic interpretation of conjugate of a partition?
For a partition $\lambda$ it is very well-known operation to take its conjugate partition $\lambda'$ which is obtained by transposing the Young diagram of $\lambda$.
A partition $\lambda$ can be ...
- 237
0
votes
0
answers
69
views
Counting covered pairs of integer partitions
I am trying to solve a problem in algebric combinatorics, where i want to prove that for integers $m,n$ and pairs ($\lambda$ , $\mu$) of integer partitions with a maximum of $m$ non zero parts which ...
- 1
2
votes
2
answers
303
views
Proof of an identity about integer partition
I'd like to know how to prove the following identity,
$$\sum_{k=1}^n k\, p(n, k) = \sum_{r,s\ge 1, rs\le n} p(n-rs)$$
where $n\in N^+$. Here, $p(n)$ counts the number of possible partitions of $n$. ...
- 93
9
votes
1
answer
746
views
Young projectors in Fulton and Harris
In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the ...
- 25.2k
2
votes
1
answer
98
views
Counting solutions to equations involving partitions
This is a problem that has come up in my research and seems to be true from numerical tests via Mathematica. It should be provable in general, but I have been unable to show it so far. It would be ...
- 1,295
1
vote
1
answer
97
views
Is there any proof of this identity?
$$\prod\limits_{i=1}^{\infty}\frac{1}{1-yx^{i}}=\sum\limits_{k=0}^{\infty}\frac{y^k x^k}{(1-x)(1-x^2)...(1-x^k)}$$
I know its combinatorial proof through "inspection",but is it true when $x,y$ are ...
- 45
9
votes
0
answers
183
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 $...
- 1,366
2
votes
0
answers
533
views
Proving Identities using Partition and Generating Function
I have a problem with these two questions:
Let $P_E(n)$ be the number of partitions of $n$ with an even number of parts, $P_O(n)$ the number of partitions of n with an odd number of parts, and $...
- 1,502
15
votes
2
answers
4k
views
Identity involving partitions of even and odd parts.
First, denote by $p_E(n)$ be the number of partitions of $n$ with an even number of parts, and let $p_O(n)$ be those with an odd number of parts. Moreover, let $p_{DO}(x)$ be the number of partitions ...
- 153