# Tagged Questions

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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### Relationship between ordered trees and integer partitions

I've found that there is a bijection between integer partitions and ordered rooted trees with roots of degree 2 or greater. The rigorous proof is complicated, but the gist of it is that you take the ...
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### Insight on an identity of partitions

There's a formula in Counting: The art of enumerative combinatorics by George E. Martin which I don't quite understand. Let $\Pi(r,n)$ be the number of partitions of $r$ into $n$ parts. If we want ...
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### Conjugate Partition and Multiset Equality

Suppose we have a partition of a number $n$, written as $(x_1, x_2, \dots , x_r)$. and its conjugate partition written as $(y_1, y_2, \dots , y_r)$ (assume that the conjugate has the same number of ...
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### Prove that a function described as below exists

$R(n,k,l)$ is defined like this : Imagine we have a set and we want to color every subset of it having $k$ elements with $n$ colors such that at the end of coloring, there exists a subset with $l$ ...
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### How many 5-element subsets A of {1, 2, … , 15} are there, the sum of whose elements is divisible by 5 [closed]

I would appreciate if somebody could help me with the following problem: Q1: How many 5-element subsets A of {1, 2, ... , 15} are there, the sum of whose elements is divisible by 5 ? Q2: How many 5-...
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### For a set with N members, what is the number of set partitions in which each subset is either of size 1 or 2? [duplicate]

I have a set with $N$ members $\{1,2, \dots, N\}$. I would like to know number of set partitions in which each subset is either of size $1$ or $2$, i.e., cardinality of each subset in the partition is ...
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### partitioning a set into subsets while considering preferences

i am looking for an algorithm to partition a set of P (p=~70) people into minimum G (G=~3) subsets/groups so that no group would have more than M (M=~30) maximum people/elements. Each person ...
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### Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
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### Elementary Set Theory ~ Partitions

I tried searching for a related thread to this, so please don't roast me too hard if one already exists. Anyways, if I have a set $A = \{a, b, c\}$ then $\{a, b, c\}$ would not be considered a ...
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### Integer partitions with distinct parts

Let $~~p(n)~~$ denote the number of all partitions of positive integer $~~n~~$ with distinct parts. I would like to find some effective algorithm for calculating $~~p(n)~~$. It seems that dynamic ...
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### How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
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### Verification of Rogers-Ramanujan identities

In Hardy's book 'Ramanujan', section 6.8 on the Rogers-Ramanujan identities, it states: None of these proofs can be called both "simple" and "straightforward", since the simplest are essentially ...
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### Splitting a set into two disjoint sets five times, minimizing pairs in the same set

Suppose you have a class of 11 students . I want to split the class into two groups five different ways, minimizing the number of times that any two students are in the same group. In more ...
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I have to proof that $p_i(n) = p_i(n − i) + p_{i−1}(n − i) + . . . + p_1(n − i).$ for every $1 \le i \le n$, where n is number of n partitions has exactly i parts. Then I have to calculate $p_5(... 0answers 26 views ### Time-scale law of Bernoulli-stopped process This post is quite long, but the problem stated carries no computational burden. Consider the equally spaced partition$t_{i}^n=\frac{i}{n}$with$i=0,...,n$of the interval$[0,1]$into$n$sub-... 1answer 17 views ### Set partition, partition into X set with Y elements. How do you partition a set into X number of new sets all that have Y elements? Example: How to partition 18 unique cards are divided to six persons, and each person gets 3 cards each. How many ... 5answers 108 views ### Does {$\Bbb Z_0$,$\Bbb Z_1$,$\Bbb Z_2 ,\cdots$,$\Bbb Z_{m-1}$} form a partition of$\Bbb Z$? "Definition 5. Let X be a nonempty set. By a partition P of X we mean a set of nonempty subsets of X such that (a) If$A, B \in \mathscr P$and$A \neq B$, then$A \cap B = \emptyset$, (b)$\bigcup\...
Say I have a partition of the set $\{1,2,3,4,5\}$. The partition is $\{\{1,3\},\{2\},\{4\},\{5\}\}$ Is there a word for set within a partition e.g. I want to say, 'one of the sets of the partition ...