All Questions
Tagged with integer-partitions combinations
52
questions
2
votes
0
answers
63
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Closed-form solution of sum over compositions?
I am interested in calculating a closed-form solution of the following sum over compositions $$ \sum_{\substack{n_1 + \dots + n_M = N \\ n_i \geq 1}} \dfrac{n_1^2 + \dots + n_M^2}{n_1(N-n_1)! \dots ...
3
votes
2
answers
577
views
How do you find the number of unique parts in a partition of an integer $n$ into $k$ parts?
Suppose I have an integer $n$ and I partition it into $k$ parts. The number of ways this can be done is given by $P(n,k)$, and it satisfies the recurrence relation:
$P(n,k) = P(n-1,k-1) + P(n-k,k)$
...
- 43
1
vote
1
answer
514
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Combinatorial arguments for number of partitions of $n$ into $k$ distinct parts
Let $Q(n, k)$ be the number of partitions of $n$ into $k$ distinct, unequal parts. Prove $Q(n + {k + 1\choose 2}, k)$ is equal to the number of ways to partition $n$ into at most $k$ parts (parts can ...
user709945
1
vote
1
answer
208
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Book Recommendations - Discrete Mathematics and Partitions of an Integer
I finished my first discrete math course this semester using mostly the excellent Kenneth Rosen (Discrete Mathematics and Applications) book that was a great help, especially in induction content and ...
- 529
0
votes
0
answers
26
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Making a group of $p$ people with $n$ available nationalities
Making a group of p people using m out of n available nationalities can be one of these two scenarios;
$m \le p \le n$ or $m \le n \le p$.
Using p,m, and n, how to evaluate the number of ways of ...
- 5,450
0
votes
1
answer
42
views
Integer Partitions of $~n~$ with restrictions.
Provide a generic formula for the number of partitions of an even number $~n~$ where one part has even value and another part also has even value.
Is there some way to approach this problem that uses ...
- 529
2
votes
2
answers
80
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Coefficient of Generating Function
Determine the coefficient of $~x^n~$ in:
$$(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}}$$
She is correct?
- 529
0
votes
1
answer
70
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Extraction of coefficient from Generating Function with partitions
Determine the coefficient of $~x ^ {15}~$ in:
$(1+𝑥^3+𝑥^6+𝑥^9+𝑥^{12}+𝑥^{15})(1+𝑥^6+𝑥^{12})(1+𝑥^9)$
How to use the fact that the desired coefficient is the number of partitions of 15 in parts ...
- 529
1
vote
1
answer
103
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Partitions of an integer with polynomials
Determine the coefficients of the polynomial $$a_0 + 𝑎_1𝑥_1 + 𝑎_2𝑥_2 + 𝑎_3𝑥_3 + ⋯ + 𝑎_𝑟𝑥_𝑟$$ that has the property that $~𝑎_𝑛 = 𝑝~$ . Where $p$ is the number of partitions of $n$ composed ...
- 529
3
votes
2
answers
241
views
Ways of distributing passengers in ships
I need help with the following combinatorial problem. There are $ K $ passengers and $ K $ ships. The passengers are denoted by $ U_1, U_2, \dots, U_K $. The objective is to find in how many ways the $...
- 778
6
votes
0
answers
141
views
Faa di Bruno's formula and alternating functions
Suppose you have a function $f(x)$ such that ${\rm sgn}\Big(\frac{d^k}{dx^k}\big(f(x)\Big) = (-1)^k$ and a function $g(x)$ such that ${\rm sgn} \Big(\frac{d^k}{dx^k}g(x)\Big) = (-1)^{(k+1)}$, $\forall ...
- 251
0
votes
1
answer
126
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What is the appropriate weight ($W_k$) (for two arbitrary partitions)?
I already asked a similar question, And from the answer I received, another question came to my mind.
A positive integer can be partitioned, for example, the number 7 can be partitioned into the ...
- 41
3
votes
1
answer
67
views
Is this true for every partitioning?
I have two categories (category1 and category2 ) and The size of both categories is equal to each other. if we partition each categories arbibtrary .Is this proposition proven? or rejected?
$n_T \...
- 41
1
vote
1
answer
125
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How many different ways to pay $2018, using only quarters, dimes, nickels, and pennies?
I have seen solutions that show how this is done for amounts such as $1. Namely I consulted this webpage's explanation--
https://www.maa.org/frank-morgans-math-chat-293-ways-to-make-change-for-a-...
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