# Questions tagged [symmetric-functions]

For questions about functions which are symmetric in its arguments.

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### Generating function of column-strict plane partitions with fixed support

A plane partition is a set $P$ of triples $(x,y,z)$ such that if in the lexicographic order we have $(x',y',z')\leq (x,y,z)$, then $(x',y',z')\in P$. We can define a partition $\lambda(P)$ that plays ...
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### Hall-Littlewood polynomials and elementary symmetric functions— Chapter III (2.8) in Macdonald's “Symmetric Functions and Hall Polynomials”

I'm confused about the proof of Chapter III (2.8), page 209 in Macdonald's book, see proof of (2.8). Here is the background. Let $\Lambda_n$ be the ring of symmetric polynomials in $r$ variables, i.e. ...
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### Is the function $f(x,y)$ describing the given graph is symmetric for any $x\neq y$? Or, only for $x=y$?

We have an infinite periodic graph with the two lengths $x$ and $y$ and we have a function $f(x,y)$ that describes a specific property of this periodic structure. Then, according to the symmetry of ...
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### Identity for the $L^1$ norm between nonnegative symmetric functions

Consider two symmetric nonnegative functions $f_1:\mathbb{R}^k \mapsto [0,\infty)$ and $f_2:\mathbb{R}^k \mapsto [0,\infty)$, i.e., for all $x \in \mathbb{R}^k$ and all permutations $\sigma(x)$ of $x$ ...
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### How to express symmetric polynomials in terms of elementary symmetric polynomials?

Firstly, I know there are very similar questions (How to express a symmetric polynomial in terms of elementary symmetric polynomials and Expressing a symmetric polynomial in terms of elementary ...
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### How does the Lipshitz norm of bounded and permutation invariant functions depend on the number of arguments?

Take a Lipshitz function on a compact subset of $R^N$ where we want to scale up $N$. If we know that (1) the functions are invariant to permutations (2) that there is a bound that is independent of ...
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### symmetric difference function

Let $X$ be a given set and let $A$ be its subset. Define $D$ as a map from the power set of $X$ to itself such that $D(B)=(A \setminus B) \cup (B\setminus A)$. I have already proved that $D$ is ...
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### Find out functions of the form $g(x,y) = \int f(x,t) f(y,t) \lambda(dt)$

I am interested in the following question. Given a symmetric function $g: \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R$ or $\mathbb R_{+}^{n}\times \mathbb R_{+}^{n} \rightarrow 0$. I am ...
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### Schur function form a basis and skew-symmetric functions

I'm trying to prove that Schur functions (in their algebraic definition) form a basis of symmetric functions, in the following way: Show that $a_{\lambda+\delta}$ form a basis of skew-symmetric ...
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### Solving for $x$ and $y$ when $(3x + y)(x + 3y)\sqrt{xy} = 14$

Solve for $x$ and $y$ when $$(3x + y)(x + 3y)\sqrt{xy} = 14$$ $$(x+y)(x^2 + 14xy + y^2) = 36.$$ I was thinking of squaring the first equation and moving on from there, but I think it'll be a bit too ...
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### Solving system of equations with fraction

I am having difficulties solving the following system : $u \neq t$ and $(t, u) \in \mathbf{R} - \{-1, 1\}$ $\frac{t}{t^2-1}-\frac{u}{u^2-1}=0$ $\:\frac{t^2}{t-1}-\frac{u^2}{u-1}=0$ I tried expanding ...
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### Ring structure on $R = \oplus_N R^n$ where $R^n:=R(S_n)$ is the vector space of class functions defined on $S_n$.

Define $R = \oplus_N R^n$ where $R^n:=R(S_n)$ is the vector space of class functions defined on $S_n$. According to Sagan, we can define a ring structure on $R$ that makes $R$ into a graded algebra, ...
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### Is this alternative proof of the inequality $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq\frac{3}{2}$ correct?

Prove that for all positive real numbers: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\geq\dfrac{3}{2}$$ This is same as this question but a different approach is used there whereas I want to verify ...
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### Question envolving bilinear forms (symmetric and skew symmetric).

Let $f: V \times V \to F$ be a bilinear form and $f$ has the follow property: Whenever $f(u,v)=0$ we have $f(v,u)=0$. The goal is to prove that $f$ is symmetric or $f$ is skew - symmetric, for that ...
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### Can every symmetric function be written as some function of a sum?

I am looking for a simple counter-example to a "theorem" about symmetric functions claimed in a published paper. The claim asserts, among many other things, that there are functions $\sigma$ ...
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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}...
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### 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 ...
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### 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 ...
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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 ...
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### 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. ...
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### 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 ...