# Questions tagged [symmetric-functions]

For questions about functions which are symmetric in its arguments.

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### prove that $\forall x \, \, \, h(a+x)=h(a-x) \Longleftrightarrow \mu=0.5$

The question is, prove that $$\forall x \, \, \, h(a+x)=h(a-x) \Longleftrightarrow \mu=0.5$$ where \begin{eqnarray} \label{eqpdf} h(x) &=&(x^2)\times \left\{ \begin{array}{cc} \int_{0.5}^{1}...
123 views

### Simplifying $\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}$

Simplify $$\frac{b^2+c^2-a^2}{(a-b)(a-c)}+\frac{c^2+a^2-b^2}{(b-c)(b-a)}+\frac{a^2+b^2-c^2}{(c-a)(c-b)}\,.$$ I tried very hard but I am not being able to solve it easily I opened up everything and ...
40 views

### Jacobi–Trudi Determinants — how to use the Lindström–Gessel–Viennot Lemma to prove the second identity?

I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur ...
32 views

### Explicit expression for certain Schur polynomials

I am trying to find explicit expression for Schur polynomials of the form \begin{equation} s_\lambda(1,q,q^2,\dots, q^{d-1},q^{-c} ,q^{d+1}, \dots,q^{N-1})~, \end{equation} with $c\in \mathbb{Z}^+$ ...
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### Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
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### Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Consider the symmetric polynomial $$P(s,t)=(s^2-1)^2+(t^2-1)^2.$$ How can we write $P$ as a polynomial in the variables $st,t+s$? The Fundamental theorem of symmetric polynomials implies this is ...
30 views

### What are the asymptotics of the $q$-binomial?

I have a rather basic question regarding the $q$-binomial $\begin{bmatrix}N \\ r \end{bmatrix}=\frac{(1-q^N)(1-q^{N-1} ) \dots (1-q^{N-r+1})}{ (1-q)(1-q^2)\dots(1-q^r) }$ as $N$ goes to infinity. ...
83 views

### Solving Functional Equation Involving Substitution

Suppose we seek a function of two variables $f(x,y)$ such that $f(x,y) = f(y,x)$ $f(x,y) = f\left(\frac{1+y}{x}, ~ y \right)$ Are there known techniques for approaching such questions? I already ...
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### Expansion of $(x+y)^n+(x+z)^n+(y+z)^n-x^n-y^n-z^n$ in terms of elementary symmetric polynomials

Consider the symmetric polynomial in $3$ variables $$f_n(x,y,z)=(x+y)^n + (x+z)^n+(y+z)^n - x^n-y^n-z^n$$ where $n\geq 0$ is an integer. I'm inquiring if there is a closed formula for the ...