All Questions
9 questions with no upvoted or accepted answers
9
votes
0
answers
275
views
pattern-avoiding permutations vs multi-core partitions
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
- 43.1k
7
votes
0
answers
152
views
Inequality of product of discrete cosines
Let $k,a,b,c$ be odd positive integers. Consider the following inequality:
$$
\sum_{x,y \in [k]} \cos^a\bigg(\frac{2\pi}{k}\cdot x\bigg) \cdot \cos^b\bigg(\frac{2\pi}{k}\cdot y\bigg) \cdot \cos^c\bigg(...
- 71
6
votes
0
answers
365
views
Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 43.1k
5
votes
0
answers
131
views
Identity for classes of plane partitions
There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...
- 43.1k
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.1k
4
votes
0
answers
205
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Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
- 43.1k
2
votes
0
answers
87
views
Reference request on Plancherel measure for partitions whose parts differing by more than $1$
Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then,
$$\sum_{\...
- 43.1k
1
vote
0
answers
90
views
Dimension of a certain space of symmetric functions: Part II
This is the second installment of my earlier MO question.
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
- 43.1k
0
votes
0
answers
171
views
Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.1k