All Questions
11 questions
2
votes
0
answers
83
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Formula for sum involving products of (symplectic) Schur functions
This question is a continuation of a question asked yesterday which had a very nice answer.
Consider the summation
$$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
- 553
6
votes
1
answer
131
views
Number of semistandard tableaux of all possible shapes fitting within some rectangle
Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum
$$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$
where $S_\lambda$ denotes the Schur ...
- 553
2
votes
0
answers
79
views
Skewed plane partition with only row fillings reversed
The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
- 51
1
vote
1
answer
308
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hook length formula for plane partitions
The hook length formula give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ...
- 568
3
votes
0
answers
144
views
Counting integer partitions below some Young diagram
Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the ...
- 956
0
votes
0
answers
91
views
Addition theorem for Schur function in multivariable
Working with the following problem Expansion in Schur function of negative binomial exponent
I need to find the expansion of
$$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$
in terms of schur ...
- 685
1
vote
1
answer
209
views
Number of paths to a specific vertex in the Young's lattice
Consider the Young's lattice. What is the number of paths starting from the origin (0) to a specific Young diagram?
For instance, the Young diagram corresponding to the integer partition 1+1+1 has 1 ...
- 153
1
vote
0
answers
69
views
LGV scheme: Any situations where the weights shift differently for each path?
In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder
In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...
- 5,474
6
votes
1
answer
469
views
Refined reverse plane partition generating function
I have a simple question about the generating function for reverse plane partitions:
$$\sum_{\pi \in RPP(\lambda)} z^{|\pi|}= \prod_{s \in \lambda} \frac{1}{1-z^{h_{\lambda}(s)}}$$
There's a ...
- 81
13
votes
1
answer
564
views
Coincidences between average Catalan tableaux
There are Catalan number $C_n$ of standard Young tableaux of shape $(n,n)$, which we view as $2\times n$ matrices. Denote by $P_n$ the average of these matrices:
$$
P_n \, := \, \frac{1}{C_n} \, \...
- 17k
6
votes
0
answers
174
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How to represent the even signed permutations by Young tableaux?
The well-known RSK correspondence established the connection between table pair (P,Q) and the permutations in symmetry group Sn(Coxeter group of type A). Also, there is a similar correspondence for ...
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