Twelvefold way

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In combinatorics, a branch of mathematics, the twelvefold way provides a unified framework for counting permutations, combinations and partitions.

Let N and X be finite sets. Let \,n = |N| and \,x = |X|. Thus N is an n-set, and X is an x-set. The general problem we consider is the enumeration of functions from N to X.

There are three different conditions which may be imposed upon f : N \rightarrow X.

  1. no condition;
  2. f is injective;
  3. f is surjective.

There are four different equivalence relations which may be imposed upon the set of functions from N to X.

  1. equality;
  2. equality up to a permutation of N;
  3. equality up to a permutation of X;
  4. equality up to permutations of N and X.

These criteria can be joined in 3 \times 4 = 12 ways.

Contents

[edit] Selection versus Partition

A function f : N \rightarrow X admits two different combinatorial interpretations.

  • f partitions N into x disjoint subsets, labeled by the elements of X.

[edit] Ordered Selections and Partitions

If f is injective, then f does not select any element more than once. In this case we say that f selects elements without repetition. If f is not required to be injective, then we say that repetition is allowed.

If f is surjective, then every element of X labels a non-empty subset of N. In this case we say that f is a partition of N into x parts. If f is not required to be surjective, then we say that f partitions N into at most x parts.

[edit] Unordered Selections

We interpret a function f : N \rightarrow X as a selection from X. The elements of X are said to be indistinguishable if we do not distinguish between two selections that differ by a permutation of N.

Formally, an unordered selection is an equivalence class of functions f : N \rightarrow X. Two functions f and F represent the same unordered selection if and only if there exists a bijection h : N \rightarrow N such that \, F = f \circ h.

Ordered selections are often called "permutations", and unordered selections are often called "combinations". The word "permutation" has a second meaning: namely, a bijection from a set to itself. We hope that this dual usage is not unduly confusing.

[edit] Unordered Partitions

We also interpret a function f : N \rightarrow X as a partition of N. The elements of N are said to be indistinguishable if we do not distinguish between two partitions that differ by a permutation of N. If the elements of N are indistinguishable, then we usually speak of partitions of n, rather than of an N-set.

Formally, an unordered partition is an equivalence class of functions f : N \rightarrow X. Two functions f and F represent the same unordered partition if and only if there exists a bijection h : X \rightarrow X so that F = h \circ f.

[edit] Balls and Boxes

Placing "balls" in "boxes" is a useful metaphor for counting problems. We think of N as the set of balls, and X as the set of boxes. The balls may be either distinguishable or indistinguishable, and the same is true of the boxes.

Injectivity of f means that no box may contain more than one ball. Surjectivity of f means that each box will contain at least one ball.

[edit] Formulas

  • Functions from N to X.
n-permutations of an x-set with repetition allowed.
\, x^n
  • Functions from N to X, up to a permutation of N .
n-combinations of an x-set with repetition allowed.
{x+n-1 \choose n}
  • Functions from N to X, up to a permutation of X
Unordered partitions of an n-set into at most x parts.
\, S(n,1) + S(n,2) + \cdots + S(n,x)
whereas \, S(n,k) is a Stirling number of the second kind.
  • Functions from N to X, up to permutations of N and X
Unordered partitions of n into at most x parts.
\, p_1(n) + p_2(n) + \cdots + p_x(n)
where \, p_k(n) is the number of partitions of n into k parts.
  • Injective functions from N to X
n-permutations of an x-set, without repetition.
(x)_n = \frac{x!}{(x-n)!}
  • Injective functions from N to X, up to a permutation of N
n-combinations of an x-set, without repetition.
{x \choose n} = \frac{x!}{(x-n)!\, n!}
  • Injective functions from N to X, up to a permutation of X.
1\ \mbox{if}\ n \leq x, \ 0\ \mbox{if}\ n>x
  • Injective functions from N to X, up to permutations of N and X.
1\ \mbox{if}\ n \leq x, \ 0\ \mbox{if}\ n>x
  • Surjective functions from N to X
Ordered partitions of an n-set into x parts.
\, x!\, S(n,x)
  • Surjective functions from N to X, up to a permutation of N.
Ordered partitions of n into x parts.
{n-1 \choose x-1}
  • Surjective functions from N to X, up to a permutation of X.
Unordered partitions of an n-set into x parts.
\, S(n,x)
  • Surjective functions from N to X, up to permutations of N and X.
Unordered partitions of n into x parts.
\, p_x(n)

[edit] Generalizations

We can generalize further by allowing other groups of permutations to act on N and X. If G is a group of permutations of N, and H is a group of permutations of X, then we count equivalence classes of functions f \colon N \rightarrow X. Two functions f and F are considered equivalent if, and only if, there exist g\in G, h \in H so that F = h \circ f \circ g. This extension leads to notions such as cyclic and dihedral permutations, as well as cyclic and dihedral partitions of numbers and sets.

[edit] References