All Questions
6
questions
0
votes
1
answer
42
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Integer Partitions of $~n~$ with restrictions.
Provide a generic formula for the number of partitions of an even number $~n~$ where one part has even value and another part also has even value.
Is there some way to approach this problem that uses ...
- 529
2
votes
2
answers
80
views
Coefficient of Generating Function
Determine the coefficient of $~x^n~$ in:
$$(x^2 + x^4 + x^6 + ... + x^{n-1})(x + x^3 + x^5 + ... + x^{n-2})$$
Where $~n~$ is an odd number.
How to describe the possible combinations of coefficients ...
- 529
1
vote
1
answer
73
views
Number of partitions of $n$ with restrictions
Find the ordinary generating function for the number of partitions of n in which all parts are odd and none surpasses 7. My answer is:
$$\prod\limits_{i=1}^7 \frac{1}{1-x^{2i}}$$
She is correct?
- 529
0
votes
1
answer
70
views
Extraction of coefficient from Generating Function with partitions
Determine the coefficient of $~x ^ {15}~$ in:
$(1+π₯^3+π₯^6+π₯^9+π₯^{12}+π₯^{15})(1+π₯^6+π₯^{12})(1+π₯^9)$
How to use the fact that the desired coefficient is the number of partitions of 15 in parts ...
- 529
1
vote
1
answer
103
views
Partitions of an integer with polynomials
Determine the coefficients of the polynomial $$a_0 + π_1π₯_1 + π_2π₯_2 + π_3π₯_3 + β― + π_ππ₯_π$$ that has the property that $~π_π = π~$ . Where $p$ is the number of partitions of $n$ composed ...
- 529
1
vote
0
answers
52
views
Relation of relative numbers of (restricted) ways to distribute identical / distinct objects into distinct bins
If want to know if the following inequality holds for general values of $s \leq n \ll m$.
$$\frac{C_0(n,m,s)}{C_0(n,m)} \leq \frac{p(n,m,s)}{m^n}$$
$C_0(n,m) = \binom{n+m-1}{m-1}$ is the number of ...
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