Mathematics Stack Exchange
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Questions tagged [multisets]

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

A multiset, also called a bag, is a generalization of a set in which elements can appear more than once. Formally, a multiset can be defined as a pair $(A, m)$ where $A$ is a set and $$m : A \to \{1, 2, 3, ...\}$$

For each $a \in A$, the number $m(a)$ is called the multiplicity of $a$, and counts how many times $a$ appears in the multiset.

Reference: Multiset.

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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 theorem is usually phrased more clumsily, without…
MJD
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Extended stars-and-bars problem(where the upper limit of the variable is bounded)

The problem of counting the solutions $(a_1,a_2,\ldots,a_n)$ with integer $a_i\geq0$ for $i\in\{1,2,\ldots,n\}$ such that $$a_1+a_2+a_3+\ldots+a_n=N$$ can be solved with a stars-and-bars argument. What is the solution if one adds the constraint that…
Niaz Mohammad Khan
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Distributing groups of objects into boxes

How can I enumerate the number of ways of distributing distinct groups of identical objects (but various cardinality) into $k$ boxes such that at most one box is empty $(1)$ and no combination of objects is repeated between boxes ? The order of…
lbeziaud
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Combinatorics: Number of possible 10-card hands from superdeck (10 times 52 cards)

I have the following problem from book "Introduction to Probability", p.32 A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, the superdeck has 52 · 10 = 520 cards, with 10…
NoBackingDown
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How to find the number of $k$-permutations of $n$ objects with $x$ types, and $r_1, r_2, r_3, \cdots , r_x$ = the number of each type of object?

How can I find the number of $k$-permutations of $n$ objects, where there are $x$ types of objects, and $r_1, r_2, r_3, \cdots , r_x$ give the number of each type of object? I'm still looking for the solution to this more general problem out of…
Cam
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Stirling numbers of the second kind on Multiset

Stirling numbers of the second kind $S(n, k)$ count the number of ways to partition a set of $n$ elements into $k$ nonempty subsets. What if there were duplicate elements in the set? That is, the set is a multiset?
shilk
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How many permutations of a multiset have a run of length k?

Background $\newcommand\ms[1]{\mathsf #1}\def\msP{\ms P}\def\msS{\ms S}\def\mfS{\mathfrak S}$Suppose I have $n$ marbles of $c$ colors, where $c≤n$. Let $n_i$ denote the number of marbles of color $i$. Let $\msP=(1^{n_1} 2^{n_2} \dots c^{n_c})$ be…
hftf
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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 $\Phi \subseteq P \times Q$ satisfying the…
Douglas S. Stones
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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 order, one counts the digits globally and, for…
Alexander Sibelius
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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 all the permutations of all the non-empty subsets of…
fede
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Odd number of reals with equal partitions

Consider the following problem: You are given a multiset (a set with repetitions allowed) of $2n+1$ real numbers, say $S = \{r_1, \dots, r_{2n+1}\}$. These numbers are such that for every $k$, the multiset $S - \{r_k\}$ can be split into two…
Aryabhata
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Number of possible sets for given N

How many possible valid collections are there for a given positive integer N given the following conditions: All the sums from 1 to N should be possible to be made by selecting some of the integers. Also this has to be done in way such that if any…
Amit
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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 struggling with the right. I do no see how the k is now on…
Pluto_lover
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How to find unique multisets of n naturals of a given domain and their numbers?

Let's say I have numbers each taken in a set $A$ of $n$ consecutive naturals, I ask myself : how can I found what are all the unique multisets, which could be created with $k$ elements of this set $A$? For example I've got $A=[1,2,\dots,499]$. If I…
Eldros
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Give a combinatorial proof for a multiset identity

I'm asked to give a combinatorial proof of the following, $\binom{\binom n2}{2}$ = 3$\binom{n}{4}$ + n$\binom{n-1}{2}$. I know $\binom{n}{k}$ = $\frac{n!}{k!(n-k)!}$ and $(\binom{n}{k}) = \binom{n+k-1}{k}$ but I'm at a loss as to what to do with the…
SharpObject
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