4

How many way can we write the integer $n$ as the sum of $k$ non-negative integers such that the largest number of the $k$ non-negative integers is $d$. The condition is the largest number in all $k$ non-negative must be $d$.

For example, I can write $10$ as $3+3+2+1+1$ and $3+2+2+2+1$ if $k = 5$ and $d = 3$.

TMM
  • 10,065
haman
  • 51
  • i have condition the k non negative parts must be smaller than or equal to d – haman Oct 01 '16 at 00:03
  • True, I retracted my close-vote. Hopefully that question can provide some ideas however. – Bobson Dugnutt Oct 01 '16 at 00:05
  • If you know the largest is $d$, then you can reduce it to the somewhat cleaner version of finding $k - 1$ numbers of at most $d$ that add up to $n - d$. After that you can do an inclusion-exclusion counting argument based on starting from the problem without the condition on $d$, and subtracting all wrong solutions, but I'm not sure if there's a nicer way to solve the remaining problem. – TMM Oct 01 '16 at 02:11
  • The title of the question says "at most $d$" but the main text of the question says "must be $d$" (i.e., equal to $d$) and the given example omits the sum $10=2+2+2+2+2$ which would be an example of a sum of integers "at most $d$". I conclude that you mean "equal to $d$", and therefore this is identical to the duplicate question. – David K Jan 06 '18 at 16:46
  • On second thought, maybe both questions should be closed as "unclear what you're asking," due to the obvious conflicts between the problem statements and the examples. (Both statements say "non-negative" but the examples use only positive numbers, excluding sums such as $10=3+3+3+1+0$.) – David K Jan 06 '18 at 16:49
  • 1
    A better duplicate source might be this question. There are probably others. – robjohn Apr 28 '20 at 03:11

0 Answers0