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I have a function which produces a polynomial in $t$ with positive integer coefficients, and I know that these factor as product of $t-$ numbers of the form $[k]_t = 1+t+...+t^{k-1}$ . Is there a way to factor the polynomials in sage which readily give the factorization in terms of the $t-$numbers?

For example, I have a polynomial $(t^2-t+1)(t^2+t+1)(t+1)^2=(t^5+t^4+t^3+t^2+t+1)(t+1)$. I want the program to return $[6]_t[2]_t$.

Thank you, this is very nice.

Writing polynomial at a product of t-numbers I have a function which produces a polynomial in $t$ with positive integer

Writing polynomial at a product of t-numbers I have a function which produces a polynomial in $t$ with positive integer

Let $p(t)\in \mathbb{Z}_{\geq 0}[t]$. Write $p(t)=\sum_i a_i t^i$. Then we say $p(t)$ is unimodal if $a_0 \leq a_1 \leq ... \leq a_k \geq a_{k+1} \geq ... \geq a_n$ i.e, the sequence of coefficients increase at first and then decrease, they don't 'jump around' ; there is no phenomenon like increase then decrease then increase again. Given such a polynomial, how can we check unimodality in sage?

Thank you very much!

From @rburing's answer... slightly modified since taking coefficients in the polynomial ring somehow only gives the non-

From @rburing's answer... slightly modified since taking coefficients in the polynomial ring somehow only gives the non-

No... I meant your earlier program didn't work if there was some internal zeroes

Actually coeffs don't take into account if some coefficient $a_k=0$ in the middle. So this works fine if there is no int

Checking whether a polynomial in unimodal Let $p(t)\in \mathbb{Z}_{\geq 0}[t]$. Write $p(t)=\sum_i a_i t^i$. Then we say

Checking whether a polynomial in unimodal Let $p(t)\in \mathbb{Z}_{\geq 0}[t]$. Write $p(t)=\sum_i a_i t^i$. Then we say

I want to expand a given polynomial in $n$ variables, homogeneous of degree $k$ as a linear combination of Non-Symmetric Macdonald polynomials $E_{\alpha}$ where $\alpha$ varies over $\mathbb{Z}^n_{\geq 0}$ with $\sum \alpha_i = k$.

Background: We know that these Macdonald polynomials do indeed form a basis of the vector space of homogeneous degree $k$ polynomials in $n$ variables. The Non-Symmetric Macdonald polynomials I am interested in is the type $GL_n$ kind. And their sage implementation can be found here: sage documentation

Bottom line is that we have a basis of a vector space already implemented in sage. Now how do we use it to compute coefficients of any vector when written in terms of this basis?

Can you explain what is happening in the repM definition please?

Non-Symmetric Macdonald expansion I want to expand a given polynomial in $n$ variables, homogeneous of degree $k$ as a l

Running a search algorithm I want to write a program to check whether a given rational function $f(x,y)$ in two variable

Defining vector partition functions in sage I want to define a vector partition function starting from a given list of v

Edited post for clarity... please check.

Understood. I wanted to get all partitions by somehow subtracting one part and knowing the smaller partitions. Do you ha

Defining vector partition functions in sage I want to define a vector partition function starting from a given list of v

Defining vector partition functions in sage I want to define a vector partition function starting from a given list of v

I want to define the following matrix in sage: $J$ is of size Partitions(n), and $J_{\lambda, \mu} = 1$ if $\lambda'=\mu$ and $0$ else. Can someone please help?

This was exactly what I was looking for. Thank you so much.

Define matrix indexed by partitions I want to define the following matrix in sage: $J$ is of size Partitions(n), and $J_

Thank you. I will try sage 9.1

Nope. Doesn't work

plot not working in windows I am using sagemath on windows 10. If I do f(x)=x plot(f) my sagemath complete

I want to define a function in a polynomial ring in several variables.

R.<x1,x3,x5>=PolynomialRing(QQ)

I am trying to define a function that takes $(i,j)$ to $x_i^j$.

I tried

def f(i,j):
   return xi^j

This does not work. I tried replacing xi with x[i], that doesn't work. Can someone please tell me what I am doing wrong and how to fix it? If instead of taking 3 variables I take only 1 variable then the method works.

I am defining my ring as R.<x>=InfinitePolynomialRing(QQ), and this should give me ring with variables x[1],x[2],... etc. right? Now I want to differentiate a polynomial with respect to x[1] variable. So I defined f=x[1]^3 (for example). I am trying f.derivative(x[1]) but that does not work. It shows " 'typeerror': argument 'var' has incorrect type (expected sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular, got InfinitePolynomial_dense)." Can someone please explain what is wrong and what I should do to fix it?