Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

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4
votes
3answers
101 views

Counting partition of set that $i$ and $i+1$ are not in one part

I have to count the number of partitions of the set $\{1,\ldots,n\}$ under the constraint that for each $i$, the elements $i$ and $i+1$ are in different parts. The my idea is: We have two situation ...
1
vote
0answers
26 views

Proof of the formula for the number of components in all partitions of a given number

I have to show that this formula is the number of components in all partitions of number $n$: $$\sum_{i=1}^{n}\sum_{j=1}^{[n/i]}\sum_{k=0}^{n-ij}A_i(k) \cdot A_{j-1}(n-ij-k)$$ $A_k(n)$ is number of ...
1
vote
1answer
163 views

If $P(n, k)$ is the number of partitions of $n$ elements into $k$ sets, then $P(n, k) = kP(n-1, k) + P(n-1, k-1)$ [closed]

A partition of the set $\{1, 2, \dots , n\}$ into $k$ parts is a way of writing the set as a disjoint union of k subsets. For example $\{1, 2, 3, 4, 5\} = \{1, 4\} \cup \{2, 3\} \cup \{5\}$ is a ...
2
votes
1answer
71 views

Explain this generating function

I have a task: Explain equation: $$\prod_{n=1}^{\infty}(1+x^nz) = 1 + \sum_{n=m=1}^{\infty}\lambda(n,m)x^nz^m $$ $\lambda(n,m)$ - is number of breakdown $n$ to $m$ different numbers (>0) It's ...
1
vote
0answers
73 views

Integer partitions without rotated solutions?

I'm searching for an algorithm to determine a list of all integer partitions of a number $n$ into a fixed number $m$ of summands (say $n=6$ and $m=4$), for instance to be stored into a list of ...
1
vote
1answer
35 views

Growth rate of ordered bounded partitions

Let $P_n(k,i)=|\{(d_1,\cdots,d_i): \ \sum_{j=1}^id_j=n, \ \forall j\ 1\leq d_j\leq k\}|$, the number of ordered partitions of $n$ into $i$ parts with individual parts bounded by $k$, with no piece ...
2
votes
0answers
47 views

Partitions of $\mathbb{R}^+$ into subset closed by sum and product

Suppose we can partition $\mathbb{R}$ into two subset $A,B$, both non empty and closed by sum and product. Let $0\in A$, and suppose that exists $b\in B$. Then $b^2\in B$. Now, $-b\in B$, cause if ...
6
votes
2answers
263 views

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) ...
31
votes
3answers
724 views

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 ...
1
vote
1answer
139 views

Count the number of ways n different-sided dice can add up to a given number

I am trying to find a way to count the number of ways n different-sided dice can add up to a given number. For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ...
1
vote
0answers
74 views

How to describe any partition a set

For ignore of a better word, I will use word "partition" try to describe what I mean. How to describe partition(where over lapping subsets are allowed) of a set mathematically? In another word, ...
3
votes
1answer
44 views

Partitioning a totally ordered set into three subsets according to the order

Consider a set $S$, and a total order $R$ over that set. Part (a) Given some element $e \in S$, explain why it is possible to partition $S$ into the following three sets: $$S_1 = \{ x \in S ...
1
vote
2answers
162 views

Geometric meaning of reflexive and symmetric relations

A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of ...
4
votes
1answer
104 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
3
votes
0answers
36 views

Arithmetic properties of the partition function

Ramanujan mentioned in his paper in 1920 that "it appears that there are no equally simple properties for any moduli involving primes other than these three" $p(5n+4)\equiv0 \mod 5$ $p(7n+5)\equiv0 ...
2
votes
1answer
65 views

partition with infinite entropy?

Let $P$ be an infinite partition of the interval $[0,1]$. Let $P$ have elements $I_i$ which has Lebesgue measure $m(I_i)$. Then the entropy of $P$ is defined by $\sum_i -m(I_i)\log m(I_i)$. Can this ...
4
votes
1answer
111 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as ...
1
vote
1answer
193 views

What is the count of the strict partitions of n in k parts not exceeding m?

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of ...
0
votes
1answer
56 views

Factorial Taxicab Number

What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, ...
3
votes
0answers
64 views

How many partitions of 12 that fit the requirements?

How many partitions of $12$ are there that have at least four parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to $4,3,2,1$? ...
2
votes
1answer
96 views

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 $n$ ...
12
votes
4answers
2k views

Prime Partition

A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
0
votes
1answer
89 views

Partitions applications in physics

Is there any direct application of all developments related to partitions? I am specially interested in physics but cryptography or other mostly theoretical areas would also be a good answer. By ...
2
votes
0answers
24 views

asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
4
votes
2answers
820 views

Keep getting generating function wrong (making change for a dollar) [duplicate]

Possible Duplicate: Making Change for a Dollar (and other number partitioning problems) I am working on the classic coin problem where I would like to calculate the number of ways to make ...
0
votes
1answer
77 views

Partition numbers with restriction on the greatest part *and* on the number of positive parts

I’m looking at partition numbers. OEIS A008284 says that the number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$, is equal to the number of partitions of $n$ into $k$ ...
2
votes
1answer
41 views

Partitioning into groups with maximal mixing

Suppose I have a class of 30 students and I want to give them 8 assignments to do in groups of 3. As far as possible I'd like the students to work with as many different students as possible. Ideally ...
1
vote
1answer
47 views

constant length of blocks in partitions

Let's assume we have partitions $P_k$ of the set $\{1,...,n\}$. If we choose two partitions it can happen, that each of them has a constant length of its blocks, but that the intersection of these two ...
0
votes
0answers
58 views

Difficulty parsing combinatorics exercise

I am working through the wonderful book Proofs and Confirmations by David Bressoud. In the section 2.2, I came across the following exercise, which has me scratching my head. (2.2.8) ~ Let ...
1
vote
1answer
61 views

Compactness of a set of partitions

The interval $[0,1]$ is partitioned to $n$ disjoint parts. Is the set of all possible partitions compact? There are several cases: A. All $n$ parts are connected intervals (possibly empty). In this ...
1
vote
0answers
37 views

Bound and parity integer-partition into fixed number of parts

Let $E(n,k)$ denote how many set of $k$ distinct non-negative integer are there such that their sum is an even number $\leq n$. Let $O(n,k)$ denote how many set of $k$ distinct non-negative integer ...
4
votes
2answers
601 views

Partitions and Bell numbers

Let $F(n)$ be the number of all partitions of $[n]$ with no singleton blocks. Find the recursive formula for the numbers $F(n)$ in terms of the numbers $F(i)$, with $i ≤ n − 1$ Find a formula for ...
4
votes
3answers
1k views

Number of cycles of all even permutations of $[n]$ and number of cycles of all odd permutations differ by $(-1)^n (n-2)!$

I'm trying to solve task 44 of the first chapter of Stanleys Enumerative Combinatorics (found here). Show that the total number of cycles of all even permutations of $[n]$ and the total number ...
1
vote
2answers
75 views

What does George Andrews mean by i in “Theory of Partitions”?

From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck): What do these last two sentences mean? I don't get "where exactly $f_l$ of the ...
0
votes
0answers
28 views

An identity relating sum of number of partitions to sum of number of parts

I encountered this identity while studying about the Kac determinants in CFT. $$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$ Here $P(N-pq)$ is the number of partitions of ...
0
votes
0answers
43 views

How to calculate least moves of fruit.

I have a question, I'll try to abstract from the real problem to not lose people. What I'm really looking for is the name of the algorithm or class of problem to find my solution. I feel that this is ...
0
votes
3answers
217 views

Ways of Distributing $n$ balls among $k$ boxes, each box containing $L \leq x_i \leq M$ or $0$ Balls

I need to calculate the number of ways of distributing $n$ balls among $k$ boxes, each box may contain no ball, but if it contains any, then it must contain $\geq L$ & $\leq M$ balls. This ...
3
votes
2answers
95 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
1
vote
1answer
62 views
0
votes
1answer
45 views

Example where a cover is fewer than a partition?

Given a Cartesian product of sets $X \times Y$, A (combinatorial) rectangle is a set $A \times B$ where $A \subseteq X$ and $B \subseteq Y$. Given a function $f : X \times Y \rightarrow \{ 0, 1\}$ one ...
0
votes
0answers
28 views

generalization of bijection proof

$p(n|\text{odd parts}) = p(n|\text{distinct parts})$ I need bijection proof and generalization of this proof. for $k = 7$ odd parts are : ...
2
votes
1answer
81 views

Integer Partition into Powers

Is there any way to count the number of integer partitions of a number N into powers of two such that each size is repeated a power of two times? Ok so the recurrence can be expressed by: $a(0)=1$, ...
3
votes
1answer
80 views

Existence of a prime partition

I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts. For instance, as given in another ...
2
votes
1answer
42 views

A bound on the balanced equipartition of a multi-set of integers

A balanced equipartition of a multi-set of $2n$ integers is a partition into two multi-sets $S_1$ and $S_2$ of size $n$ such that the sum of the integers in $S_1$ is as close as possible as the sum of ...
1
vote
0answers
61 views

Generation of n terms that sums to a specific value?

I'm about to conduct an integer program where I am to transport some products from different cities to docks and refinement stations. I am however required to work with simulated data where I only ...
2
votes
1answer
34 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
4
votes
7answers
485 views

Book recommendation for Integer partitions and q series

I have been studying number theory for a little while now, and I would like to learn about integer partitions and q series, but I have never studied anything in the field of combinatorics, so are ...
3
votes
1answer
126 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
2
votes
0answers
25 views

What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?

Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with ...
1
vote
2answers
253 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...