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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21
votes
0answers
514 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
17
votes
0answers
400 views

Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
12
votes
0answers
139 views

Partition of ${1, 2, … , n}$ into subsets with equal sums.

The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988). Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers ...
9
votes
0answers
80 views

Closed form for $\sum_{n=1}^\infty \frac{1}{P(n)}$, where $P(n)$ is the partition function.

Is there a closed form for the following infinite series? $$\sum_{n=1}^\infty \frac{1}{P(n)}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$$ where $P(n)$ is the partition function.
8
votes
0answers
82 views

Partitioning points with a line

Let $A_{m, n} = \{1, 2, \dots, n\} \times \{1, 2, \dots, m\}$. A straight line would partition the points into two sets. How many ways are there to do it? Let $p_{m, n}$ be that number. Apparently ...
7
votes
0answers
180 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
7
votes
0answers
269 views

Partitions of 13 and 14 into either four or five smaller integers

There are exactly 18 partitions of the integer 13 into 4 parts, as on the left of the table, and also 18 partitions into 5 parts, as on the right of the table: $$\begin{array}{c|c} 10+1+1+1 & ...
7
votes
0answers
269 views

Is there some sort of correspondence between groups and partitions of a set?

Every group action on a set $S$ partitions the set into orbits. Conversely, for every partition of $S$ is there a group action such that the set of orbits of the group action equals the partition? ...
7
votes
0answers
354 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any ...
7
votes
0answers
284 views

Is there a connection between the 3D random walk constant and the partition function?

In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk, ...
6
votes
0answers
233 views

Invariant element in the tensor product of rectangular Specht modules?

Denote by $\mathfrak{S}_k$ the symmetric group on $k$ elements. Let $\lambda=(n^2\times n)=(n^2,\ldots,n^2)$ be a rectangular partition and $k=n^3$. Denote by $S_\lambda$ the Specht module ...
5
votes
0answers
109 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
5
votes
0answers
130 views

Sum over subsets of a multiset

I have a sum that looks like the following for some multiset $S$ and some function $f$ of $n$ variables which does not depend on the ordering of its arguments: $$\sum_{\{k_1,\dots, k_n\}\subset ...
5
votes
0answers
58 views

Degree distribution of a graph of integer partitions

Consider a graph whose nodes are the integer partitions of $n$, connected by an edge if you can get from one partition to the other by ‘sliding one square of its Young diagram’ – that is, if one ...
5
votes
0answers
331 views

Which unordered partition of $n$ gives rise to the largest number of ordered partitions?

A quick look at the wikipedia article on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim ...
5
votes
0answers
125 views

On Applications of the Murnaghan-Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnaghan-Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
4
votes
0answers
42 views

Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$

I am currently taking discrete math and have been given the following question to answer. Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$ where ...
4
votes
0answers
77 views

The number of partitions by distinct positive numbers

Let $N>0$ be a natural number and let $P(N)$ denote the number of ways to write $N$ as a finite sum of $a_i$ such that the $a_i$ are strictly decreasing positive natural numbers. There is a paper ...
4
votes
0answers
137 views

A general Combinatorics problem (Coefficients of the q factorial)

I was solving a combinatorics problem when I encountered difficulties. The problem was: $x_1 \in \{0,1\}$ $x_2 \in \{0,1,2\}$ . . $x_{n-1}\in\{0,1,2..,n-1\}$ We have to find the number of ways ...
4
votes
0answers
115 views

Why limit Euler's Partition function P to $k\leq\sqrt n$ instead of $k\leq n$?

I solved a Project Euler problem (I won't say which one) involving the Partition Function P. I used equation #11 from the above link: $$P(n) = \sum_{k=1}^n (-1)^{k+1}\bigg(P\Big(n-{1\over ...
3
votes
0answers
39 views

Amount of combinations of sets summing to number

(Apologies for the confused arbitrariness here; I don't have experience in formal maths to make abstract my lay-person thoughts, but I've tried my best.) I have $x$ identical but order-important sets ...
3
votes
0answers
33 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
3
votes
0answers
53 views

Problem for number theory

Here it is: $c , n \in \mathbb{N}$ and $x_1,x_2,\ldots,x_n \in \mathbb{N}\cup \{0\}$ $c= 1 x_1 + 2 x_2 + \ldots + n x_n$ How many solutions $\{x_1,x_2,\ldots,x_n\}$ are there? I do not know number ...
3
votes
0answers
140 views

Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's ...
3
votes
0answers
57 views

Counting the number of partitions that are a distance d away from a fixed partition.

Given a positive integer $N \in \mathbb{Z}^{\geq 0}$ let $Partitions(N)$ denote the set of all partitions of $N$, where a tuple $\left(f_1,\ldots,f_N \right)$ is a partition of $N$ if $\sum_{i=1}^N ...
3
votes
0answers
41 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 ...
3
votes
0answers
86 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 $4,3,2,1$? ...
3
votes
0answers
98 views

Asymptotics of partitions in at most n parts, bounded by r

For every positive integers $n,r,w$ define $$ p_w(n,r)=\#\{ (i_1,...,i_r) | \, 0\leq i_1 \leq \dots \leq i_r\leq n, \, i_1+\dots+i_r=w\} $$ as the number of partitions of $w$ in at most $r$ piece ...
3
votes
0answers
72 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
3
votes
0answers
132 views

If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$

Question: If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$ with integrator $\alpha$. ...
2
votes
0answers
32 views

Known classic problem or not?

There is a set of positive whole numbers without null. I have to find the minimal number of subsets of the original set so, that the the sum of two numbers in a subset can't be the value of a number ...
2
votes
0answers
43 views

Which are partitions are for $\mathbb{Z}$? Which are covers for $\mathbb{Z}$? Which are both or neither?

(a) {{x : x is an even integer}, {x : x is an odd integer}} (b) {{x : x is an even integer}, {x : x is divisible by 3}} Note - divisible by 3 includes negative integers, yes? -3, -6, -9, ... and ...
2
votes
0answers
128 views

Partition counting problem with cap on pairwise intersection

Fix $T_1,\ldots T_m$ as pair-wise disjoint $k$-subsets of $\{1,2,\ldots,m\cdot k\}$, so that $|T_i|=k$ and $|T_i\cap T_\ell|=0$. For any $j\le k$, how many sets of the form $\{C_1,\ldots,C_m\}$ are ...
2
votes
0answers
40 views

What are the combinatorial numbers appearing in these repeated derivatives?

Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$. I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$. Let $\lambda$ be a partition of $n$, i.e. a tuple of ...
2
votes
0answers
50 views

Notation for the set of all integer partitions

I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. ...
2
votes
0answers
49 views

how to count possible planar bipartitions?

i want to find out what small fraction of a solution space a metaheuristic search is actually covering. this case comes down to the number of possible bipartitions for a non-bipartite, undirected ...
2
votes
0answers
48 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 ...
2
votes
0answers
28 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 ...
2
votes
0answers
67 views

Partitions and divisor functions: what is known about their relations?

If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$): \begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of ...
2
votes
0answers
37 views

Quantiles when the population contains only one unique value

My apologies for the mixing of the terms quartile and quantile below. I am interested in the general case of quantiles, but I'm using a quartile as a specific example. Also feel free to clarify any ...
2
votes
0answers
69 views

What is the least integer of additive dimension 4?

Say that $m$ is the additive dimension of $n\in\Bbb N$, and write $m=\operatorname{ad}n$, if $m$ is the greatest integer for which there is an irredundant $m$-element set $M\subset\Bbb N$ that ...
2
votes
0answers
32 views

Canonical partition function identity

The grand canonical partition function, Z, is given by $$ Z = \sum_{M=0}^\infty G(M)z^M =\frac{V_0(z)U_L(z)}{1−U(z)V (z)} \tag{1} $$ where G(M) is the canonical partition function of a chain of ...
2
votes
0answers
75 views

Distributing problem using generating functions

For $r\in\mathbb{Z^*}$, let $a_r$ denote the number of ways to distribute $r$ identical objects into $3$ identical boxes, $b_r$ be the number of distributions so that the boxes are to be non-empty, ...
2
votes
0answers
44 views

explicit random cake cutting

I like to split a given interval, let's say $[0,1]$, randomly to a given number $n$ parts. A random input may be provided, like for example a sequence of random numbers $\omega=(r1, r2, ...
2
votes
0answers
103 views

Bell number with minimum bound on partition size

I know that the Bell number $B_n$ is the number of ways to partition a set of $n$ elements into distinct non-empty subsets. Is there a variant of this number that specifies the minimum number of ...
2
votes
0answers
21 views

Terminology for breaking partition diagram into “L”'s

When one thinks about partitions, it's quite normal to consider pieces of the partition diagram, such as rows, columns, arms, legs, hooks, etc. One decomposition of particular interest to me is ...
2
votes
0answers
105 views

What is the correct technical term for this generalization of an integer partition?

Given a vector $v$ with non-negative integer coordinates, is there a technical term for an unordered tuple of vectors $(v_1,\dots, v_k)$ with non-negative integer coordinates such that $v_1+\dots+v_k ...
2
votes
0answers
169 views

Partition minimizing maximum of Euler's totient function across terms

Given natural numbers $M$ and $N$, I'd like to find a partition of $2^N$ with $M$ or fewer terms, $t_1 + t_2 + ... + t_M$, such that $\max(\phi(t_1), \phi(t_2), ..., \phi(t_M))$ is minimized, where ...
2
votes
0answers
263 views

Generating Function of Integer Partition Such that at Least One Part is Even

I've been having a few issues coming up with a generating function for an integer partition such that at least one part is even. What I have got so far is: The generating function with no ...
2
votes
0answers
37 views

order of elements in a partition using Maple

I determined this whole partition but I just want to have the finer the partition for example: I have this ...