Intereting Posts

differential equations, diagonalizable matrix
The Impossible puzzle (“Now I know your product”)
Flip a coin until a head comes up. Why is “infinitely many tails” an event we need to consider?
Topology proof: dense sets and no trivial intersection
Is there matrix representation of the line graph operator?
What is the fractional constant of integration?
Dense and locally compact subset of a Hausdorff space is open
Coupon collector's problem using inclusion-exclusion
Infinite recurrence relation which depends on subsequent sequence values
Difficulty evaluating complex integral
What is Mazzola's “Topos of Music” about?
prove that $\tan(\alpha+\beta) = 2ac/(a^2-c^2)$
Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n – 1} \le n$
How to show the Symmetric Group $S_4$ has no elements of order $6$.
The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

I want to count the number of necklaces, with $n$ beads in total, where the alphabet of beads is $\{1,\ldots,k\}$, and where the number of beads with color $i$ is $n_i$. For example, if $n=4$, and $n_1 = n_2 = 2$, the following necklaces are the only possible

$$1122$$

$$1212$$

I have been able to find formulas for different variations of the problem, but not this one. If there is no nice formula, a generating function is always a nice compromise.

- Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
- Product of binomial coefficient as a basis
- Sum of multiplication of all combination of m element from an array of n elements
- Find the number of arrangements of $k \mbox{ }1'$s, $k \mbox{ }2'$s, $\cdots, k \mbox{ }n'$s - total $kn$ cards.
- Probability of people attending same and different places
- Number of $m\times n$ $(0,1)$ matrix with given row sums and unique columns

- Why a complete graph has $\frac{n(n-1)}{2}$ edges?
- Improvised Question: Combination of selection of pens
- What is the minimum number of locks on the cabinet that would satisfy these conditions?
- Combination with repetitions.
- In the card game Set, what's the probability of a Set existing in n cards?
- Number of combinations and permutations of letters
- Total number of unordered pairs of disjoint subsets of S
- Positive integers less than 1000 without repeated digits
- How to geometrically show that there are $4$ $S_3$ subgroups in $S_4$?
- How Many Symmetric Relations on a Finite Set?

There is a reasonably nice generating function coming from the Pólya enumeration theorem. It is given explicitly in, for example, this blog post: the number you want is the coefficient of $\prod r_i^{n_i}$ in

$$\frac{1}{n} \sum_{d | n} (r_1^{n/d} + … + r_k^{n/d})^d \varphi \left( \frac{n}{d} \right).$$

- Number of all labeled, unordered rooted trees with $n$ vertices and $k$ leaves.
- Prove that $\mathrm{Res}=\frac{f(z_0)}{g'(z_0)}$
- Does the Jordan curve theorem apply to non-closed curves?
- Integration using residues
- A simple bijection between $\mathbb{R}$ and $\mathbb{R}^4$ or $\mathbb{R}^n$?
- Finding the sum of series $\sum_{n=0}^∞ \frac{2^n + 3^n}{6^n}$
- Using the distributivity law for propositional logic
- Prove that $\vert\sin(x)\sin(2x)\sin(2^2x)\cdots\sin(2^nx)\vert < \left(\frac{\sqrt{3}}{2}\right)^n$
- Is it sufficient to check weak convergence on a (weak* or strongly) dense subset of the dual?
- Odds of winning at minesweeper with perfect play
- An Identity Concerning the Riemann Zeta Function
- Kan extensions for linear categories
- Proving irreducibility of $x^6-72$
- Show that the function is discontinuous in $\mathbb{R}$
- Intuition surrounding units in $R$