# Tagged Questions

22 views

37 views

### Is the number of associative $n$-ary algebraic operations on a finite set with 2 cardinality always 8?

We know that if $n = 2$ then the operation is called a binary operation. $\circ$ on set $X$ is a function $\circ : X \times X \rightarrow X$. And the number of all associative binary operation on a ...
58 views

### Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?

So I'm stuck on this problem. If you perform a faro out-shuffle (i.e. a perfect "riffle shuffle" where the top and bottom cards stays in place) on a pack of 52 cards ($n=26$), you can get back the ...
40 views

29 views

### $PGL_d(F)$ is 2-transitive but not 3-transitive if $d > 2$

An exercise asks to prove that: If $d > 2$, then the projective general linear group $PGL_d(F)$ of dimension $d$ over a field $F$ is 2-transitive but not 3-transitive on the set of points of the ...
73 views

### Using Plancherel's Theorem to Prove the Gauss Sum

I'm interested in proving the following: Where $p$ is an odd prime and $z$ is a primitive $p$th root of unity, we let $Q(p)=\sum^{p−1}_{k=0}z^{k^2}$. Prove: $|Q(p)|=\sqrt{p}$. Specifically, I want ...
74 views

### Math competitions resource at university level

I want some problems especially in Algebra field for math competitions at undergraduate math students level. Does anybody here know book, website,... that I can use?!
48 views

### Does the Cayley digraph $C$= [(12)(34),(123):$A_4$] have a Hamiltonian Circuit?

This is a problem I'm working on for a friend of mine. I haven't been able to solve it, or make much progress. I have drawn the digraph, and it consists of four directed cycles of three vertices all ...
63 views

### Find the number of irreducible polynomials in any given degree

For any prime $p$ find the number of monic irreducible polynomials of degree $2$ over $\mathbb Z_p$. Do the same problem for degree $3$. Generalize the above statement to higher degree ...
80 views

### Application of group theory to combinatorics

Let k be a positive integer. In how many ways one can color the edges of an equilateral triangle using k colors (two coloring schemes are considered the same if one can be obtained from the other via ...
73 views

### Recurrence for the number of $\sigma \in S_n$ with cycle length at most $r$

I have just learned that the formula is right, but the definition of $c_n^{(r)}$ was wrong. The correct problem is: Prove $$c_{n+1}^{(r)} = \sum_{k=n-r+1}^n n^{\underline{n-k}} c_k^{(r)}$$ where ...
99 views

### How many monoids of order three are there?

http://oeis.org/A058129 In the above link we can see the answer is 7. I have tried counting these and don't get 7. I am not sure what I am doing wrong so could someone go through counting these step ...
60 views

### Two generating functions are combined, how are the coefficients for the combined generating function determined?

I have a function or operation $f(x)$ that takes integer inputs $0\leq x_\mathrm{in}<2^N$ to a limited set of integer outputs $x_\mathrm{out}\in[-N,N]$. I know that if all inputs are computed then ...
152 views

### Fibonacci polynomials and factorization redux

I recently asked a question about factorizing a certain expression involving Fibonacci and Lucas polynomials. That question had a very simple and nice answer, but now I've come across another similar ...
58 views

### A factorization problem involving Fibonacci and Lucas Polynomials

Consider a sequence of polynomial $\{w_n(x)|\, n\geq 0\}$ which are defined recursively by $w_n(x)=xw_{n-1}(x)+w_{n-2}(x)$. With $w_0(x)=0$ and $w_1(x)=1$, one gets the so-called Fibonacci polynomials ...
61 views

### Proving that if $n\times n$ Hadamard matrix exists, then 4 divides $n$

Im looking for an explanation of the following: a standard way to prove that, if there exists Hadamard matrix of dimension $n > 2$, then $4|n$, is to suppose that without loss of generality every ...
25 views

148 views

### 2D Rubik's cube?

There is a $3\times3$ matrix filled by numbers 1~9 that might look like this $$\begin{bmatrix}3 & 8 & 2 \\ 4 & 1 & 6 \\ 7 & 5 & 9\end{bmatrix}$$ All its rows and columns can ...
32 views

### Intuition of Newton's identities, is it worth persuing these thoughts? Should be able to show it from special case.

I've "accidentally" stumbled on a special case of it (i=n) and it is very intuitive. Then there's some sort of "transpose" going on with the terms. So I'm thinking there might be something there. I ...
38 views

### Understanding “avoiding sequence”

Each subsequence (91674, 91675, 91672) is called a copy, instance, or occurrence of σ. Since the permutation π = 391867452 contains no increasing subsequence of length four, π avoids 1234. ...
24 views

### I suspect that a polynomial that is symmetric is a polynomial in “the sums of powers” - using a loose def of polynomial.

This is something I suspect, by this I mean I can't see it being wrong and I have evidence to suggest it is true. It may be a named theorem but given I've been exploring this with a notebook at a ...
39 views

### A bijective transform that cycles. Help with definitions requested

In many ways I am a novice with mathematics. My background is college algebra. I am attempting to write my first maths paper and am faced with sifting through mathematics I am not familiar with. It ...
39 views

EDIT (along the lines suggested by @sea turtles): Given a prime power $q$ and a positive integer $x\gt q$, how many subsets $A\subseteq{\bf F}_2^q$ have size $x$ and contain a subset $B\subseteq A$ ...
81 views

### An algebraic proof that a set has $2^n$ subsets. (I'm looking for an inductive argument.)

There will be duplicates of this, so let me explain why I am asking: I have become blind to what it may be, so I want hints. I am blind because I can do it "combinatorially". The question wishes me ...
126 views

### How are lopsided binomials (eg $\binom{n}{n+1})?$ defined?

For instance is $\binom{n}{n+1}=0$ always or something else?
45 views

### Finding the coefficient of a specific term in a polynomial without expanding/simplifying?

I'm learning about Polya's Theorem which deals a lot with the coefficients of terms in a polynomial. In one problem I had the following polynomial, $(b+w)^{16}$, and the problem basically reduces to ...
147 views

### If $|A| > \frac{|G|}{2}$ then $AA = G$

I'v found this proposition. If $G$ is a finite group , $A \subset G$ a subset and $|A| > \frac{|G|}{2}$ then $AA = G$. Why this is true ?
Suppose you have an operator $T$ on a vector space $F^n$, and you're given the minimal polynomial $m_T(x)=p_1(x)^{a_1}\cdots p_k(x)^{a_k}$, where $\sum a_i=\deg(m_T(x))=d\leq n$. Is there a general ...