For questions about or related to multisets, a notion similar to sets with the difference that elements can be repeated.

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4
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2answers
64 views

Using Sticks and Stones for Counting number of Ways

From the first twenty positive integers, how many ways can we select 6 integers so that no two integers from the six chosen ones are consecutive? I tried using sticks and stones, but my thought ...
-1
votes
0answers
21 views

What would the cardinality be if you took difference into account?

If you have a set A which has elements: {a,b,c,d,e} and set B which has elements: {f,g,h,j} then would the cardinality of (AB)/(BA) be 20 or would it be zero ?
0
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1answer
39 views

Combinatorial proof (multi choose)

I'm struggling to explain why these two sides are equal in a non algebraic way. Basically I'm looking for a combinatorial proof of why these sides are equal. I know they are equal by algebra. N ...
-2
votes
0answers
12 views

Find similarity metric for multi-sets

I try to find a similarity metric for multi-sets which outputs the same as the Jaccard-similarity in the case that both sets aren't multisets. Jaccard-similarity: $C_1, C_2$ are sets. ...
4
votes
1answer
140 views

Unfamiliar notation in an AoPS paper

Here it is, from this paper: Proposition 5.1.1. The number of skyline polyominoes of area $A$ and width $w$ is $\left(\!\binom{w}{A-W}\!\right) = \binom{A-1}{w-1}$. I'm referring to the first ...
0
votes
1answer
15 views

$r-$ combinations of multisets

I have the topic $r-$ combinations of multisets in my notes Let $S=\{\infty a_1,\infty a_2,\ldots ,\infty a_k\}$ , then the formula for the $r-$*combinations of S* is given as: ...
0
votes
2answers
23 views

Probability with Multiple Sets and Anti Sets

Calculate: $P(A \cap B'\cap C')$ Given: $P(A) = 0.7$ $P(B) = 0.8$ $P(C) = 0.75$ $P(A \cup B) = 0.85$ $P(A \cup C) = 0.9$ $P(B \cup C) = 0.95$ $P(A \cup B \cup C) = 0.98$ I can upload a pic of my ...
0
votes
3answers
31 views

Combinatorial problem - multisets

As I am solving some basic combinatorial problems today, I found out this problem: How many different 5-digit numbers can be formed from digits 2, 2, 7, 7, 9? Can someone guide me to a solution for ...
1
vote
1answer
31 views

Solving basic combinatorics

I started course in combinatorics and, as I'm still not much into it, I'm solving some basic problems to start with. So here is one of them: How many 5-digit positive integers are there such that 9 ...
0
votes
1answer
39 views

Structural Induction Subsets

Consider the set $S \subset \mathbb{N}^2$ of ordered pairs of integers defined by the following recursive definition: • $(3, 2) \in S$ (basis) • If $(x, y) \in S$, then $(3x − 2y, x) \in S$ ...
1
vote
1answer
27 views

Maximum product for multisets with same sum

Given a positive number N, among all multisets (containing only positive numbers) with sum N, is there a reliable method for determining the set with the maximum product? For example, for N = 5, the ...
0
votes
2answers
53 views

Permutations of a Multi-Set

Find the number of permutations of the multi-set {m.1,n.2}, where m,n $\in N $, which must contain m 1's. I thought the permutation is $\frac{(m+n)!}{m!n!}$ since multi-set is basically a collection ...
1
vote
0answers
33 views

Question concerning defining a particular class of functions

I have a multiset of real numbers $X \subseteq \mathbb{R} $ and I want to create a class of injective function to map the elements of $X$ to the unit interval(so basically a normalization). However ...
4
votes
1answer
28 views

Maximum operation order for a set of integers

Say we are given the positive integers $[1,1,2,2,3]$ We want to know what the maximum number is using only the operators $+$, $\times$. For this set the maximum operation is ...
0
votes
0answers
28 views

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. ...
1
vote
0answers
12 views

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 ...
1
vote
1answer
107 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: ...
0
votes
0answers
5 views

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 ! ...
2
votes
1answer
39 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: $$ ...
1
vote
1answer
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 ...
0
votes
1answer
17 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 ...
1
vote
0answers
28 views

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
1
vote
1answer
40 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 ...
0
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1answer
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 ...
0
votes
1answer
38 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 ...
0
votes
0answers
48 views

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' ...
1
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1answer
93 views

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 ...
0
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0answers
22 views

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 ...
0
votes
1answer
47 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 ...
0
votes
0answers
54 views

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 ...
0
votes
2answers
94 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.
0
votes
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 ...
1
vote
1answer
85 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 ...
0
votes
1answer
71 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 ...
0
votes
3answers
272 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 ...
2
votes
0answers
51 views

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 ...
2
votes
1answer
82 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 ...
2
votes
3answers
74 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?
2
votes
2answers
349 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 ...
1
vote
0answers
67 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 $X=\{1,1,2,3,3\}$ ...
1
vote
2answers
107 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 ...
3
votes
3answers
149 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?
0
votes
2answers
60 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 $a$ ...
7
votes
2answers
273 views

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 ...
0
votes
1answer
90 views

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 ...
1
vote
0answers
73 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 ...
1
vote
1answer
164 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 ...
1
vote
1answer
229 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 ...
2
votes
1answer
327 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 ...
7
votes
0answers
156 views

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 ...