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9 votes
2 answers
344 views

Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices

Let $G_{m,n}$ denote the number of $m\times n$ $(0,1)$-matrices that avoid the submatrix $\bigl({1\atop1}{1\atop0}\bigr)$. (Submatrices need not be contiguous.) Here are some small values (not yet on ...
ho boon suan's user avatar
6 votes
0 answers
207 views

Parameter independence of Stanley's "content formula". Why?

For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively. R. Stanley remarked following ...
T. Amdeberhan's user avatar
3 votes
1 answer
272 views

Generating function for parity in hooks

Let $\lambda\vdash n$ denote an integer partition of $n$ and $\frak{H}_{\lambda}$ be the multiset of hook lengths of $\lambda$. Further, let $o(\lambda)=\#$ of odd entries and $e(\lambda)=\#$ of even ...
T. Amdeberhan's user avatar
3 votes
1 answer
281 views

Analytic expression for the coefficient of a multivariate polynomial

Does there exist some method for finding an analytic expression for the coefficient of $z_1^kz_2^kz_3^k$ in: $$[(1+z_1)(1+z_2)(1+z_3)(1+z_1z_2)(1+z_1z_3)(1+z_2z_3)(1+z_1z_2z_3)]^{k}$$ or is it ...
Fabius Wiesner's user avatar
3 votes
0 answers
312 views

Enumerating a class of polynomials

How many equivalence classes of $\Bbb F_2[x,y]$ polynomials with $x$ degree $n_x$ and $y$ degree $n_y$ are there such that each $y^i$ coefficient (polynomial in $\Bbb Z[x]$) is distinct and $x^i$ ...
user avatar
2 votes
1 answer
310 views

Generating function for A300483 (related to Chebyshev polynomial of first kind)

Let $a(n)$ be A300483. Here $$ a(n) = 2\int\limits_{t \geqslant 0}T_n\left(\frac{t+1}{2}\right)\exp(-t)\,dt. $$ where $T_n(x)$ is $n$-th Chebyshev polynomial of first kind. Let $b(n)$ be an integer ...
Notamathematician's user avatar
2 votes
0 answers
117 views

A multi-variable "Fibonacci polynomial"?

There is a tremendous literature on the Fibonacci sequence, including its polynomial analogue $F_{-1}=0, F_0=1$ and $$F_n(x)=xF_{n-1}(x)+F_{n-2} \qquad \text{for $n\geq1$}.$$ What I have found is the ...
T. Amdeberhan's user avatar
1 vote
1 answer
233 views

Closed-form formula for a multivariate polynomial

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and for $k\geq 2$ let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\...
Dragan Stevanovic's user avatar
1 vote
0 answers
101 views

Construct generating functions of series of palindromic polynomials

I have a problem that is generating a series ($d=2,4,\ldots,20,\ldots$) of pairs of $4 d$-degree palindromic (self-reciprocal) polynomials. The first three members ($d=2,4,6$) of the first pair are: \...
Paul B. Slater's user avatar
0 votes
1 answer
60 views

Formula for $P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \mathbb{N}_+} \left( \prod_{i=1}^m k_i^{a_i} \right) $

Let $\mathbb{N}_+ = \{ 1, 2, \dots\} $. For a given sequence of elements $\{a_i \}_{1 \leq i \leq m} $in $ \mathbb{N}_+ $, we define \begin{equation} P(d) = \sum_{\sum_{i=1}^m k_i = d, k_i \in \...
zhjzwlys's user avatar
0 votes
1 answer
250 views

If the coefficient of the polynomial positive

I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$ $$\bar{S}(k)=\...
GGT's user avatar
  • 685
0 votes
0 answers
84 views

Arithmetic triangles and unimodality of its rows

Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence. How to prove that the coefficients form an ...
Mikhail Gaichenkov's user avatar