All Questions
11 questions with no upvoted or accepted answers
7
votes
0
answers
174
views
A diagonal generating function for Fibonacci: Part II
In my earlier MO question, I mentioned although we have for the Fibonacci numbers that
$$F_n=[x^n]\left(\frac1{1-x-x^2}\right),$$
is there a function $F(x)$ such that $F_n=[x^n]\left(F(x)\right)^n$?
...
- 43.2k
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 ...
- 43.2k
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
3
votes
0
answers
315
views
When does the Taylor coefficient of $e^{\sin x}$ vanish?
If $f(x)=\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\frac{a_3}{3!}x^3+\frac{a_4}{4!}x^4+\cdots$ is an exponential generating function for $\{a_k\}_{k\geq1}$ then
$$e^{f(x)}=1+\frac{a_1}{1!}x+\frac{a_1^2+a_2}{2!...
- 43.2k
3
votes
0
answers
123
views
$q$-series for the number of rectangles in a square lattice
Given a partition $\lambda\vdash n$ of $n$, look at its Young diagram $Y_{\lambda}$. Let $a(\lambda)$ be the number of squares (of all sizes) in $Y_{\lambda}$. For example, if $n=4$ then $a(4)=4, a(3,...
- 43.2k
2
votes
0
answers
110
views
Asking for a generating function for an arithmetic sequence
For fixed integer $n\geq1$, let $c_m(n)$ be the number of divisors $d$ of $m$ such that $n<d\leq 2n$. Here is an experimental generating function for which I ask:
QUESTION. Is this true?
$$\sum_{m\...
- 43.2k
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 ...
- 43.2k
2
votes
0
answers
98
views
Two-variable generating functions over coprime pairs
I am studying a sequence $(\alpha_{p,q})$ indexed by a pair of coprime integers; this sequence arises naturally in the study of a particular set of spaces in geometric topology, but unfortunately the ...
1
vote
0
answers
43
views
Correspondence between monomer-dimer heaps and words in 2 alphabets
For background and some illustrative pictures, refer to this preprint by A M Grasia and G Ganzberger: Fibonacci polynomials. For the present purpose, it suffices to read into pages 4 and 5.
The part ...
- 43.2k
1
vote
0
answers
134
views
Counting unions of unlabelled connected graphs
My question can be stated as follows: let $X$ be a hereditary family of unlabelled graphs closed under disjoint unions. Suppose we know, for each $n$, the number $c_n$ of connected graphs in X on $n$ ...
- 183
1
vote
0
answers
95
views
Generating series of free PROs
Let
\begin{equation}
G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q)
\end{equation}
be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...
- 437