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Let $\mathbb N_n^*=\{1,2,3,,...,n\}$ be the set of integers greater than zero. How many increasing maps from $\mathbb N_n^*$ to $\mathbb N_p^*$ do exist ?

It is not mentioned that $p\ge n$, So i think increasing means monotonically increasing (and not strictly).

I tried for $n=4$, $p=3$ and find out:

If we call the function $f$ and $f(1)=3$ then $f(i)=3$ for $i=2,3,4$ is the only possibility

if $f(1)=2$ then $f(i)=2$ for $i=2,3,4$ is possible, but not only this $\{2,2,2,3\}$ $\{2,2,3,3\}$ $\{2,3,3,3\}$ are also allowed.

It is like, if we have a $n\times p$ grid and we're in the last row and the column number depends on $f(1)$ and we have to do $n-1$ steps either right or upwards. I get as result:

Call $m:=min\{n-1,p-1\}$

$\sum\limits_{i=0}^m\binom{n-1}{i}+\sum\limits_{i=0}^{m-1}\binom{n-1}{i}+...+\sum\limits_{i=0}^0\binom{n-1}{i}$

Can you verify this ? Thanks in advance (Sorry for my english).

OBDA
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1 Answers1

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I don't understand how you got your last formula, but don't you find it a bit strange that the number does not increase when $p$ is increasing? For example, set $n=1,p=2$.

Let me suggest another interpretation of the problem: Suppose you have an increasing map $f:\Bbb N_n^* \to \Bbb N_p^*$. Then you can reconstruct the map from the images of $f$ given with multiplicity. So there is a one-to-one correspondence of increasing maps $f:\Bbb N_n^* \to \Bbb N_p^*$ to ways of choosing $n$ numbers out of $p$ with replacement.

benh
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  • 1.part Yes, i see the mistake, that's because in my example p is smaller than n, so the upper limit of the sum decreases, but in general not. 2.part so you choose n out of p with replacement that means $p^n$ possibilities. and you can always bring it in a ''right order'' ? – OBDA Mar 11 '14 at 15:39
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    You choose $n$ out of $p$ with replacement, but it is about combinations, not permutations. See here http://math.stackexchange.com/questions/474741/formula-for-combinations-with-replacement and here http://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition – benh Mar 11 '14 at 16:16
  • clear now, thanks. – OBDA Mar 11 '14 at 16:47