All Questions
Tagged with young-tableaux integer-partitions
22
questions
0
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7
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Conjugate of a Gel'fand pattern
Background:
A Gel'fand pattern is a set of numbers
$$
\left[\begin{array}{}
\lambda_{1,n} & & \lambda_{2,n} & & & \dots & & & \lambda_{n-1,n}...
- 248
2
votes
0
answers
79
views
Weighted sum over integer partitions involving hook lengths
I am trying to compute the following quantity:
$$ g_n(x) = \sum_{\lambda \vdash n} \prod_{h \in \mathcal{H}(\lambda)} \frac{1}{h^2} \exp\left[x\sum_i \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\right]...
- 484
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0
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148
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Question on definition of Schur polynomial from Fulton's Young Tableaux.
In the following from Young Tableaux by Fulton, what happens if our Tableaux has numbers greater than $m$? Fulton gives an example with $m=6$, but according to his definition, we also have a monomial ...
- 12.1k
3
votes
1
answer
68
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Parity of hooklengths in a partition diagram
Main Question
Let $\lambda\vdash n$ be a partition, with hooklengths $\{h_1,\dots,h_n\}$ in its partition diagram. Is there a formula for determining
$$\#\{h_i\text{ even}\}-\#\{h_i\text{ odd}\}?$$
...
- 405
1
vote
1
answer
173
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Is a Standard Tableau determined by its descent set?
Suppose $\lambda\vdash n$ is a partition. Associated with this partition is the set of Standard Young Tableau $\text{SYT}(\lambda)$ such that the associated Young Diagram is filled in with the numbers ...
- 405
0
votes
1
answer
92
views
Partition of integer and its conjugate
For the partition $(6,4,4,2)$ of integer $16$, if we draw its Young diagram with four rows of boxes, one below the other, of size $6$, $4$, $4$, and $2$, then flipping the resulting Young diagram ...
- 3,065
1
vote
1
answer
152
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Distribution of number of terms in integer partitions
SOLVED: This is the Gumbel distribution
Let $\pi^n_i$ be set containing the terms in the $i$-th integer partition of the natural number $n$, according to whatever enumeration. For example, for $n = 5$ ...
- 11
5
votes
2
answers
229
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Given a Ferrers diagram, prove that $\det(M)=1$
Let $\lambda$ be a Ferrers diagram corresponding to some
integer partition of $k$. We number the rows and the columns, so that the
j'th leftmost box in the i'th upmost row is denoted as $(i,j)$. Let
$...
- 273
1
vote
0
answers
88
views
Maximum value, function of partition and its conjugate
Suppose that we have $n\in \mathbb{Z}_{+}$ and some $\alpha\ge 3$.
I am trying to find maximum value of:
$\sum_{i,j=1}^{n}|\lambda_{i}-\lambda_{j}^{*}|^{\alpha},$
over
$\{\lambda\in \mathbb{Z}^...
user617199
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43
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Smallest rectangle inscribing Young tableaux
I am interested in knowing the name of any of this characteristics of a Young tableaux (Ferrer's diagram): Smallest rectangle that contains it or the area of such a rectangle. For instance:
Partition ...
- 623
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
0
answers
260
views
Counting Semistandard Young Tableaux For Triangular Shapes?
If $k \leq n$ I denote the Young diagram with shape $(n,n-1,n-2,\ldots,1)$ by $\lambda^{n,n-1,\ldots,1}$. I write $f^{\lambda_n^{n,n-1,\ldots,1}}$ to count the number of semistandard Young tableaux ...
- 3,788
0
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1
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505
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Counting the number of semistandard Young Tableaux with maximum entry $n.$ Reference/Formula request
Question: If $k \leq n$ let $\lambda_k$ be a Young diagram with square $k \times k$ shape. I write $\#_{\lambda_{k}^n}$ to count the
number of semistandard Young tableaux with shape $\lambda_k$ and
...
- 3,788
1
vote
0
answers
71
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Why is the ladder tableau of an $e$-restricted partition $e$-restricted?
Lemma 3.40 on page 46 in Mathas's "Iwahori-Hecke Algebras and the Symmetric Group" states
Suppose that $\lambda$ is an $e$-restricted partition of $n$. Then the ladder tableau $\mathfrak{l}_e^\...
- 997
9
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0
answers
183
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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