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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25
votes
0answers
806 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
415 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 ...
9
votes
0answers
89 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.
9
votes
0answers
84 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 $...
8
votes
0answers
308 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 & 9+1+...
8
votes
0answers
366 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
279 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
285 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, $$\frac3{(2\pi)^3}\...
6
votes
0answers
74 views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
6
votes
0answers
242 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
125 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
148 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 S}f(...
5
votes
0answers
61 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
357 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
136 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
100 views

Almost-Linear Sequence of Positive Integers Excluding a Near-Quadratic Integer Sequence

I remember that I have seen a similar problem to this one here: The set of natural numbers that don't belong to a set (which is a duplicate of $m$ doesn't come in the sequence $a_n=[n+\sqrt{n}+...
4
votes
0answers
69 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 $R=\{(0,0),(0,4),(...
4
votes
0answers
212 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 proof....
4
votes
0answers
79 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
127 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 $4,3,2,1$? ...
4
votes
0answers
140 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
117 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 2}(3k-1)\...
3
votes
0answers
48 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
38 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
58 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
42 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
107 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
82 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
137 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 $\alpha$. Quasi-...
2
votes
0answers
25 views

enumerating all permutations of partitions

Given a set of $n$ items, I would like to enumerate all the permutations of all the partitions of those $n$ items. For example, in the case of $n=3$ (with Bell number 5), there are 13 permutations of ...
2
votes
0answers
49 views

Partitioning a set of integers (with Alice and Bob)

Let $ d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} $ (not necessarily distinct) be given. Define $ D:=\operatorname{lcm}(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $. (1) Alice claims that whenever $ \...
2
votes
0answers
47 views

The number of partitions of $n$…Subbarao

(Subbarao) The number of partitions of $n$ in which each part appears two, three, or five times equals the number of partitions of $n$ into parts congruent to $2, 3, 6, 9,$ or $10$ modulo $12$. ...
2
votes
0answers
38 views

How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - $$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - f(...
2
votes
0answers
35 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
2
votes
0answers
56 views

$ \forall n \in N$ edges of $K_{2n+1}$ can be partitioned to hamiltonian cycles.

A hamiltonian cycle is a cycle which visits each vertex of the graph exactly once. A hamiltonian path is a path which visits each vertex of the graph exactly once. We need to prove that: 1-...
2
votes
0answers
26 views

show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd

I would like to show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd, or preferably even more generally that $\sum_{j=0}^n (-1)^j {n \brack j }_q =\frac{1}{2}((-1)^n+1)(q;q)_{\frac{n}{2}}$. Using ...
2
votes
0answers
33 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
46 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
137 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
44 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
56 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
54 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 $-b\...
2
votes
0answers
29 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 $n\...
2
votes
0answers
75 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 }n,\notag\\...
2
votes
0answers
39 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
85 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
33 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
85 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, ...