Intereting Posts

On the $j_!$ of a sheaf
Proving $(1 + 1/n)^{n+1} \gt e$
Difference between $\mathbb{Q}/(X-1) \otimes_\mathbb{Q} \mathbb{Q}/(X+1)$ and $\mathbb{Q}/(X-1)\otimes_{\mathbb{Q}}\mathbb{Q}/(X+1)$?
Understanding the definition of Cauchy sequence
Least rational prime which is composite in $\mathbb{Z}$?
Proof that $q^2$ is indivisible by 3 if $q$ is indivisible by 3.
Proving formula for sum of squares with binomial coefficient
Number of real roots of $\sum_{k=0}^{n}\frac{x^{k}}{k!}=0$
What is the importance of Calculus in today's Mathematics?
If $a(n)=n^2+1$ then $\gcd(a_n,2^{d(a_n)})=1\text{ or }2$?
dimension of a subspace spanned by two subspaces
Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely
Why is gradient the direction of steepest ascent?
Direct formula for a variation of Josephus problem:
Why do some people place the differential at the beginning of their integral?

How many monotonically increasing functions are there with domain $\mathbb Z_7$ and codomain $\mathbb Z_5$?

So, the domain is $\{0, 1, 2, 3, 4, 5, 6\}$ and the codomain $\{0, 1, 2, 3, 4\}$.

I found the total number of functions to be $405$ and know that it will be a recurrence relation, but I have had no luck deriving it.

- Does there exist a set of exactly five positive integers such that the sum of any three distinct elements is prime?
- Can the Identity Map be a repeated composition one other function?
- Pigeonhole principle application
- Combining kindergardeners in 'fair' cookie-baking groups. Kirkman's schoolgirl problem extended version
- $f(f(f(x))) = x$. Prove or disprove that f is the identity function
- Fold, Gather, Cut
- Non-Decreasing Digits
- Infinite sum involving ascending powers
- $f'(c) \ge 0 , \forall c \in (a,b)$ then $f$ is increasing in $$ , proof of this without Mean Value theorem
- Different of mapsto and right arrow

None of the functions can be strictly increasing, so we count what I would rather call the monotonically non-decreasing functions.

Such a function can be completely described once we know **how many** of $0,1,2,\dots,6$ are mapped to $0$, how many are mapped to $1$, how many are mapped to $2$, and so on. For if $x_0$ of them are mapped to $0$, it must be the **first** $x_0$ of $0,1,2,\dots$. And if $x_1$ of them are mapped to $1$, it must be the *next* $x_1$ of $0,1,2,\dots$, and so on.

Here is a quick shorthand description of such a function: $10402$. This says that $1$ object in $\mathbb{Z}_7$ is mapped to $0$, $0$ objects are mapped to $1$, $4$ objects are mapped to $2$, none are mapped to $3$, and $2$ are mapped to $4$. In function language, we have $f(0)=0$, $f(1)=f(2)=f(3)=f(4)=2$, and $f(5)=f(6)=4$.

So we want to count the number of solutions of the equation $x_0+x_1+x_2+x_3+x_4=7$ in non-negative integers. By *Stars and Bars*, this number is $\binom{7+5-1}{5-1}$.

- Non combinatorial proof of formula for $n^n$?
- Eigenvalues of product of a matrix and a diagonal matrix
- Drawing by lifting pencil from paper can still beget continuous function.
- The connection between quantifier elimination, $\omega$-categorical and ultrahomogenous
- About the first positive root of $\sum_{k=1}^n\tan(kx)=0$
- What makes Probability so difficult to get it right in the first place?
- Is there a simpler way to falsify this?
- Factorial Summation Definition
- Show $\psi$ and $\Delta$ are identifiable
- Don't understand casting out nines
- Cardinality of the set of bijective functions on $\mathbb{N}$?
- Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
- Show how to calculate the Riemann zeta function for the first non-trivial zero
- Is this a Correct Proof of the Principle of Complete Induction for Natural Numbers in ZF?
- distribution of categorical product (conjunction) over coproduct (disjunction)