All Questions
25 questions
26
votes
1
answer
2k
views
Is the derivative of $x^n + x^{n-1} + \dots + x + 1$ irreducible?
I am working on some combinatorics problems. One of my problems leads to the following question:
Is it true that the derivative of $x^n + x^{n-1} + \dots + x + 1,$ namely $nx^{n-1} + (n-1)x^{n-2} + \...
- 431
18
votes
2
answers
2k
views
Can Schwartz-Zippel be formulated for commutative rings instead of fields?
The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for $(1+x^n)=1+x^n(\operatorname{...
- 2,464
17
votes
1
answer
687
views
Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...
- 823
13
votes
1
answer
228
views
Recognizing algebraic independence among Schur polynomials
Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
- 85.6k
11
votes
1
answer
722
views
A closed formula for $A_n(X)=\sum\limits_{i=0}^n X^{i^2}$
I want to know if there exists a closed formula for sum $A_n(X)=\sum \limits_{i=0}^n X^{i^2}$.
I have found if $n$ is odd then $(X^n+1)\text{ | } A_n(X)$, but I don't have found a closed formula.
- 4,074
11
votes
1
answer
339
views
Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions
Slicing cones in various ways with a plane generates conic sections identified geometrically as hyperbolas, parabolas, or ellipses and algebraically, when suitably rotated, as certain rescaled ...
- 10.5k
10
votes
2
answers
820
views
Simple question about polynomials
Starting from a problem in combinatorics, I ended up with a very simple problem about polynomials, which, unfortunately, I am not able to solve.
Say we work over $\mathbb C$. Fix $d>1$.
Is it ...
- 153
8
votes
2
answers
2k
views
What generalizes symmetric polynomials to other finite groups?
Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...
- 781
7
votes
0
answers
344
views
Irreducibility of a palindromic polynomial
I have strong reasons to believe that the palindromic polynomial $p_n(x)$ defined by
$$p_n(x) = x^{2n}+2x^{2n-1}+3x^{2n-2}+ \cdots+ nx^{n+1}+(n+3)x^{n}+nx^{n-1}+\cdots+2x+1$$
is irreducible in $\...
6
votes
1
answer
242
views
$(q,t)$-Fibonacci polynomials: area & bounce statistics
This is related to my earlier (unanswered) MO post. Preserve notations from there.
We take advantage of the one-to-one correspondence between the $(s,s+1)$-core partitions and $(s,s+1)$-Dyck paths. ...
- 43.2k
5
votes
1
answer
289
views
Cataland: Facets and partition polynomials of cluster complexes
Figure 25 on pg. 101 of "Cataland: Why the Fuss?" by Christian Stump, Hugh Thomas, and Nathan Williams depicts cluster complexes (CCs) associated with generalized $(m)$-Narayana / ...
- 10.5k
4
votes
1
answer
370
views
Determining when quotient of a polynomial ring is a Gorenstein ring
I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
- 214
4
votes
0
answers
128
views
Inequality for $q$-binomials
Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...
- 43.2k
3
votes
0
answers
151
views
Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
- 10.5k
3
votes
0
answers
107
views
Non-tree models of Lagrange inversion polynomials
The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...
- 10.5k
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
2
votes
1
answer
162
views
Maximally independent polynomial families with row symmetry
Introduction:
In the 1-dimensional case, given $m$-variables
$$\mathbf{x} = (x_1,x_2,\dots,x_m)^T,$$
the elementary symmetric polynomials $(e_k(\mathbf{x}))_{k=1}^m$ give a "symmetric basis",...
- 473
2
votes
0
answers
73
views
An iterative formula for the Kreweras-Voiculescu polynomials (reference request)
Let
$$N(x) = 1 + \sum_{k \ge 1} N_k(h_1,h_2,...,h_k) \;x^k$$
$$ = 1 + h_1 x + (h_1^2 + h_2) x^2 + (h_1^3 + 3h_1h_2 + h_3)x^3 + (h_1^4 + 6 h_2 h_1^2 + 4 h_3 h_1 + 2 h_2^2 + h_4) x^4 + \cdots$$
be the ...
- 10.5k
2
votes
0
answers
68
views
Sampling theorems for partition polynomials (associahedra, noncrossing partitions / parking functions)
Define the associahedra partition polynomial
$$
\begin{split}
A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\
& \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) ...
- 10.5k
1
vote
1
answer
243
views
Combinatorics and geometry underlying a refined Pascal matrix/Newton identities
The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for ...
- 10.5k
1
vote
0
answers
63
views
Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion
I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...
- 10.5k
1
vote
0
answers
329
views
Outlier absences of monomials in a group of inversion partition polynomials
Revamped and updated on Sep 12, 2022:
Given the complex coefficients $a_n$ of some generic formal power, Taylor, Laurent or other series, say the ordinary generating functions (o.g.f.) $f(z) = z +a_1 ...
- 10.5k
1
vote
0
answers
89
views
Combinatorial models of the refined inverse Eulerian numbers
If I evaluate substitution of an infinite set of indeterminates $(c_1,c_2,c_3,\cdots)$ into the infinite set of refined Eulerian polynomials $[E]$ of OEIS A145271, I obtain the Taylor series ...
- 10.5k
1
vote
1
answer
177
views
Refinement of face vectors of the simplicial noncrossing hypertree complexes of McCammond
Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition ...
- 10.5k
0
votes
1
answer
349
views
Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k