For questions about or related to multisets.

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Series of multi sets

Given that a set of numbers $K = \{n_1, n_2, n_3, ... \}$. Multiple subsets are formed by randomly extracted numbers from $K$. Then series are formed by extracting numbers from the subset orderly. ...
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distance metric between multisets

I am trying to define a distance $F(X,Y)$ between two multisets $X$ and $Y$. For each pair of $x \in X , y \in Y$ there exists a distance function $f(x,y)$ which takes the range of $[0,1]$. An ...
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1answer
63 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: ...
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Weighted ordering of subgroups

As you may have guessed by my title, I've got no background in maths, but I think this is the right bit of StackExchange to ask in and hopefully I've picked the right tags... but forgive me if not ! ...
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1answer
33 views

What is Vandermonde's formula with multisets?

I need Vandermonde's formula in multi-set form. I modified the original formula but I get a mess with too many letters everywhere, is there a nice representation? Here's the original: $$ ...
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26 views

Name for a generalized relation to be a multiset?

A relation between two sets $A$ and $B$ is a subset of $A \times B$. If taking a multiset subset of $A \times B$, e.g. allowing $(a,b)$ appears twice in the subset, is there a name for such a ...
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1answer
16 views

Is there any special name for an algebraic structure (set, equivalence relation)?

I've seen the term "real multiset" but it doesn't seem to be very appropriate so i wonder whether there are any others. My second question is about multisets. In most sources i've seen one of the ...
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Notation to count tuples

I have the following set: V={(0,0),(1,2),(1,3),(2,2),(3,2)} I need to count tuples containing x = 1. Can I use |V_{(1,y)}| ? Or I should use |{(x,y) \in V | x = 1}| ? Thanks, Luiz
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1answer
27 views

Number of k-colorings as a fraction of all possible ways to color a graph

I have a graph with $n$ verices, and I want to compute the number of ways to color the graph (with no adjacent vertices having the same color) using anywhere between $1$ and $n$ colors. This number ...
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49 views

Help With sets!

Can someone help me solve this question please?? Pretend you are writing traffic accident software and want to categorize accidents by the day of the week on which they occur. Pretend there are n ...
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1answer
31 views

Multiset: notation for size and number of unique items

Given a multiset, e. g. S = {1, 1, 2, 3, 4, 4, 5}, what would be a short, concise notation to express the number of unique items in the multiset? (five in the given ...
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Number of possible multisets of multisubsets (with constraints)

Let's say I have a multiset $s = \{1, 2, 3, 4, 5\}$ (in this example there are no repeats, but there could be for any arbitrary s) Here are some examples of what I mean by 'multiset of multisubsets' ...
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Solving problem involving intersection of sets

In a certain office,one-quarter of the staff are left-handed. One-twelfth of them are left-handed and short-sighted; 13 are short sighted while 17 are neither left handed nor short-sighted.Find the ...
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X is to trees, is what multi-relations are to multi-sets

Multi-sets are like sets in which elements can be repeated. Formally, a multi-set over a set $S$ is a function $S \to \mathbb{N}$. Multi-set addition, difference, and intersection can be defined ...
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46 views

How to represent a multiset?

I have a graph $G(V,E)$ and each $v_i\in V$ has a value $v_i\cdot s$ ($v_i\cdot s$'s are not unique). How can I show a multiset representing the $v_i\cdot s$'s? This is what I have come up with so far ...
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How to find the smallest element of a “hidden” multiset?

(I am posting this question for a friend.) We are given a multiset S of, say, 1000 positive integers. Our task is to find the smallest element of S. The problem is, S is hidden. Instead, we ...
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Possible operations on multisets.

A multiset is a collection of objects with size s and weight w. For example, the collection: {a, b, a, c, a, d, e} has size = 5 and weight = 7. Possible permutations of a multiset = weight! / (unique ...
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92 views

Two sets given : solve $A \cup X = B$

Do you know how to solve this problem? I have two sets and need to solve $A \cup X = B. $ Thanks a lot for your help.
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1answer
77 views

Exterior square of multiset in representation theory

General Setting: In a paper I'm working on, the author uses multisets to describe the representation theory of the cyclic group $G = C_n = <\sigma>$ of ...
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1answer
80 views

Select r items from a set with multiplicity k and total items n.

Let N be a set of n distinct objects having the same multiplicity k. For instance, N={1,1,2,2,3,3} where n=3 and k=2. Now I want to select r numbers from ...
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1answer
69 views

Inclusion of sets

let $F$ be a multi application :$F: X\leadsto Y$ I have this definitions "If $B\subset Y$ is an non empty set we have : $F^{-1}(B)=\lbrace x\in X, F(x)\subset B\rbrace $ $F^{-1}_+(B)=\lbrace x\in ...
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226 views

Ways to select donuts

Wanted to share this puzzle: A restaurant offers choice of six different types of donuts, each available in unlimited quantity. How many ways can you select three donuts? You can pick any number of ...
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Which pmf is appropriate for this set-based problem?

Consider a finite multiset: $S = \{ \{a, a\}, \{\}, \{ a, c, d \}, \ldots, \{ a, t, u \}\}$ where $\forall s \in S$, $0 \leqslant \|s\| \leqslant n$, and where each multiset $s$ contains only ...
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1answer
80 views

Multisets with Exact Number of Repeated Integers

Given a multiset that contains 5 numbers where the numbers are from 0 to 5 inclusive, and the numbers can be repeated: a) In how many ways can you have a multiset with exactly four 4s? b) In how ...
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72 views

A set with members allowed to appear more than once

I'm looking for a definition for a set which its members could be appeared more than once! for example: $$D=\{1,1,2,4,6,6\}$$ Could we call this a set?
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343 views

Names of different kinds of sets based on uniqueness, ordering, and length

I have some descriptions of some things like sets, based on some properties, and I'm trying to find out what they're called. In the course of investigating the concepts behind Ruby arrays, I noticed ...
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58 views

Terminology for matrix whose rows are permutations of a given multiset.

Let $X=\{a_{1},a_{2},\ldots,a_{m}\}$ be a multiset. Is there a name for an $n\times m$ matrix $A$ such that the entries of each row of $A$ are equal to the set $X$. For example, if $X=\{1,1,2,3,3\}$ ...
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2answers
103 views

Proper way to define this multiset operator that does a pseudo-intersection?

it's been a while since I've done anything with set theory and I'm trying to find a way to describe a certain operator. Let's say I have two multisets: $A = \{1,1,2,3,4\}$ $B = \{1,5,6,7\}$ How ...
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3answers
139 views

Is there such a thing as a multiset with a “negative” number of some element?

Is it possible for a multiset to have a "negative" number of one or more elements? If so, how are such multisets defined, and what terminology exists for them?
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58 views

How to make precise the notion of “the multiset of roots of a polynomial function”?

A (real) polynomial function can be defined as a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a sequence $a : \mathbb{N} \rightarrow \mathbb{R}$ such that the terms of $a$ ...
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Combinatorial proof for identity $\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$ (multiset coefficients)

In class we have recently started using combinatorial proofs. I have tried this problem that our teacher has assigned as a "challenge". I understand how to receive the left hand side, but am ...
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Which definitions of builder notation exist for multiset theory?

Interesting cases would be $A=[1,1], B=[2]$ $[(a,b) \mid a \in A \wedge b \in B] = [(1,2),(1,2)]$ ? or $C=[1,2,3]$ $[x \mid c \in C \wedge x = c \mod 2] = [1,0,1]$ ? The only kind of informal ...
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68 views

Set-valued map (measurability)

I have this exercice and i want to know how to solve it : 1)- Let $X,Y$ two separable metric spaces ,let $(\Omega, \mathcal{A})$ be a measurable space ,and $f: \Omega \rightarrow X$ a measurable ...
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1answer
155 views

Difference between inclusion-exclusion problems

There are two situations.The first is a bakery which has three type of doughnuts, {6*chocolate , 6*cinnamon, 3*plain}. How many options do they have for a box of 12 doughnuts? The second question is ...
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1answer
188 views

Probability of an item being selected in a multiset

I would like to know that probability of an item occurring in a multiset (a combination of selections with repetitions). Given a set $S = \{x_1,x_2,...,x_n\}$ the number of possible unordered subsets ...
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1answer
261 views

Combinatorics/Multisets problem question

I wonder how a problem of the following type can be solved. I have looked for a solution but I am not to identify the kind of problem I am facing. I would like to know if there is a close formula or ...
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A variation on the Look and Say Sequence and some questions about it.

For information on the sequence mentioned in the title, see http://en.wikipedia.org/wiki/Look-and-say_sequence. This is an original problem. Suppose instead of "describing" the numbers in a string in ...
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4answers
487 views

Combinations of multisets - the theory?

I've read over the theory countless times, and I still have no idea how to think of it. The formula for the combinations of multisets is $C(k + r - 1, r)$, where $k$ = the number of distinct ...
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1answer
161 views

Put N items into M groups such that the sums of the N's in every M are as even as possible

I have a multiset, S, that contains N items that I wish to place into ...
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1answer
766 views

Probability of predicting, then throwing, a particular multiset for 5 dice.

My friend shared with me a story that after losing to his SO at Yahtzee, before they put the game away he just randomly predicted he would roll four 5's and a 1. He then got that roll and freaked out. ...
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1answer
257 views

Number combinations multi set

Here is my question: Consider a multiset $\{n\cdot a, n\cdot b, 1,2,3, \ldots, n+1\}$ of size $3n+1$. Determine the number of $n$-combinations. I know from my textbook that if you have a ...
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1answer
191 views

Are lexicographic and multiset the only extensions of sets that preserve well-foundedness?

It is well known that for a given set $S$ with well-founded order relation $R$, the lexicographic order that extends $R$ on tuples of $S$ is also well-founded. Also, the multiset order on the ...
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Why does mathematical convention deal so ineptly with multisets?

Many statements of mathematics are phrased most naturally in terms of multisets. For example: Every positive integer can be uniquely expressed as the product of a multiset of primes. But this ...
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203 views

What is the notation for the multisets of R?

The set of sets of elements of $R$, also known as powerset of $R$ can be typeset $2^R$. I am now interested in the set of multisets of elements in $R$. How is it called? Is there a standard notation?
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462 views

Number of k-combinations of a multiset

Say that I'm buying cakes for a party. I wish to buy $k$ cakes, and there are $n$ different kinds of cake, but only $m_i$ of each kind of cake (where $i$ denotes the $i$th kind). How many different ...
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210 views

Derangements of multisets

Find the number of ways string of numbers (may contain similar items) could be deranged so that a number is not placed in the same place as it or its similar numbers were placed. For example, ...
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3answers
386 views

Multiset Combination in Combinatorics

I want to buy a $k$-combination of doughnuts, where $k$ is any amount less than or equal to the total doughnuts available. At the bakery there are $n$ different types of doughnuts but there are ...
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313 views

permutations of a multiset having symbols with fixed multiplicity

Let $N$ be a multiset of $n$ distinct objects having the same multiplicity $k$. For instance, $N=\{a,\,a,\,b,\,b\}$ where $n=2$ and $k=2$. I was looking for the problem of counting the number of ...
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What is the number of bijections between two multisets?

Let $P$ and $Q$ be two finite multisets of the same cardinality $n$. Question: How many bijections are there from $P$ to $Q$? I will define a bijection between $P$ and $Q$ as a multiset $\Phi ...
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Probability of throwing the same multiset twice in a row with six dice

Six dice are thrown. The six dice are thrown a second time. What is the probability of getting the same numbers as in the first throw? If the order of the six numbers matters, the problem is easy, but ...