All Questions
Tagged with integer-partitions fibonacci-numbers
10
questions
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143
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Generating function of ordered odd partitions of $n$.
Let the number of ordered partitions of $n$ with odd parts be $f(n)$. Find the generating function $f(n)$ .
My try : For $n=1$ we have $f(1)=1$, for $n=2$, $f(2)=1$, for $n=3$, $f(3)=2$, for $n=4$, $...
user1137804
13
votes
2
answers
2k
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Number of ways to represent any N as sum of odd numbers? [duplicate]
I was solving some basic Math Coding Problem and found that For any number $N$, the number of ways to express $N$ as sum of Odd Numbers is $Fib[N]$ where $Fib$ is Fibonnaci , I don't have a valid ...
- 512
1
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77
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bijections between sets
Let $P_n$ be the set of compositions of $n$ where each part is at least $2, Q_n$ be the set of compositions of $n$ where each part is odd, and $R_n$ be the set of compositions where each part is $1$ ...
user747916
1
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1
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371
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Partition Function and Fibonacci Number
I am asked to prove that $$p(n) < F_n, \qquad n \geq 5$$ where $p(n), \ F_n$ are the number number of partitions of $n$ and the $n^{th}$ Fibonacci number, respectively. This is easy if you can ...
- 337
1
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0
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76
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Counting number of integer solutions to $a_1 + a_2 + a_3 + \ldots = n$ where all $a$'s must be in certain range
For a given $(n,m,k)$..
Using values in the range $(0..k)$, how many different $m$-combos exist which sum to n?
ex. for $(n,m,k)$ = $(3,3,2)$, there are 7 possible combinations. For $(5,4,2)$ ...
- 11
0
votes
2
answers
681
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Partition function and Fibonacci n-th number upperbound
i need to proof this upperbound:
"Call $p(n)$ the number of partitions of n (integer) and $F_n$ the $n-th$ Fibonacci number. Show that
$p(n)\leq p(n-1) + p(n-2)$ and that $p(n)\leq F_n$ for $n\...
- 11
2
votes
1
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495
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Covering a rectangle of size $n\times1$ with dominos
A rectangle of size $n\times1$ is given.
(a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$?
(b) In how many ways the rectangle can be covered with ...
- 16k
2
votes
1
answer
576
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The number of partitions of $n$ and the $n$th Fibonacci number.
I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . .
Let $P_n$ be the number of partitions of $n$ ...
- 45.8k
36
votes
3
answers
2k
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Very curious properties of ordered partitions relating to Fibonacci numbers
I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon.
We call an ordered ...
- 361
6
votes
2
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578
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Is an algebraic formula for the number of cyclic compositions of n known?
From Wikipedia:
In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) (...
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