Conventions fixed on this page, and what counting is deliberately not done here

This item is the page's ledger: every convention the page fixes, with the item that fixes it, and a statement of what is deliberately left to later pages.

$\mathbb{N}$ contains $0$. Every index range on this page starts at $0$, so $\sum_{k<n}$ runs over $k = 0, \dots, n-1$ and $\sum_{k<n+1}$ over $k = 0, \dots, n$. A cardinality may be $0$, a part of a composition may be $0$, and a claim true only from the second index onwards is false as stated. Three items exist only because of this: the alternating row sum of $\sum_{k<n+1}\binom{n}{k} = 2^{n}$, and $\sum_{k<n+1}(-1)^{k}\iota\!\binom{n}{k} = 0$ for $n \ge 1$ carries the hypothesis $n \ge 1$, For $m \ge 1$ the number of weak compositions of $n$ into $m$ parts is $\binom{n+m-1}{m-1}$, and the number of compositions is $\binom{n-1}{m-1}$ for $n \ge 1$ carries $m \ge 1$, and the composition count carries $n \ge 1$.

$\lvert A\rvert$ is defined for finite $A$ only, and is a natural number. The cardinality $\lvert A\rvert$ of a finite set fixes this, and what makes it well posed is claim 3 of the pigeonhole principle. It is not a cardinal number: Cardinal (initial ordinal) and cardinality ↗ is a later and different object and nothing here uses it or any cardinal arithmetic.

The empty sum is $0$, the empty product is $1$, $0! = 1$ and $0^{0} = 1$ are one convention. Each is the base clause of a recursion carried out on this page, in Finite sums and finite products of natural numbers, $\sum_{k<n} a_k$ and $\prod_{k<n} a_k$ in $\mathbb{N}$, The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$ and Exponentiation of natural numbers, $m^{n}$, and its agreement with the integer power in $\mathbb{R}$ respectively, and each agrees with the already-published Finite sums and finite products, by recursion and Integer powers $a^m$. Nothing was imported and nothing was stipulated twice.

Counts live in $\mathbb{N}$; identities involving subtraction or division live in $\mathbb{R}$. A natural number is a von Neumann natural, hence a set, and is not an element of $\mathbb{R}$; it enters through the canonical natural $\iota$. That is why the binomial theorem's coefficient is $\iota\binom{n}{k}$, and why Laws of finite sums and products in $\mathbb{N}$, and $\iota\big(\sum_{k<n} a_k\big) = \sum_{k<n} \iota(a_k)$ proves that $\iota$ commutes with finite sums and products and is injective. Injectivity is the licence to prove an identity between counts by proving it in $\mathbb{R}$.

$\binom{n}{k}$ is a count, and integrality is a theorem. The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ defines it as $\lvert [n]^{k}\rvert$, so it is a natural number by construction; that it also equals $n!/(k!(n-k)!)$ is $\binom{n}{k}\,k!\,(n-k)! = n!$ for $k \le n$; hence $\binom{n}{k}\,k! = n^{\underline{k}}$, the quotient $n!/(k!(n-k)!)$ is a natural number, and $\binom{n}{k} = \binom{n}{n-k}$. The same holds for the multinomial coefficient (The multinomial coefficient $\binom{n}{k_0,\dots,k_{m-1}}$ as the number of ordered partitions of an $n$-set into blocks of prescribed sizes).

Three notions of finite sum will coexist in the library, and each is introduced with a bridge to the previous one: over an initial segment (Finite sums and finite products, by recursion), over a finite index set (The sum $\sum_{i \in S} a_i$ over a finite index set, and its product form, well posed because of A finite sum is unchanged by a permutation of its index range: $\sum_{k<n} a_{\pi(k)} = \sum_{k<n} a_k$ for every bijection $\pi : n \to n$), and in an arbitrary monoid (The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity ↗, later in the reading order). The bridges are what stop them drifting into three unrelated notions.

Truncated difference. On this page $n - m$ always means the unique $j$ with $m + j = n$ when $m \le n$, and $0$ otherwise. No negative number is ever formed, and each statement true only under $m \le n$ says so.

These are statements about the reading order, not about the library as a whole.

• Inclusion and exclusion, the systematic repair of a count whose blocks overlap. The sum rule needs disjointness, and the companion page shows what goes wrong without it; the correction term belongs to the next page of this track.
• The number of surjections, and the counting of set partitions and of unordered partitions of an integer. All are natural sequels to the material here and none is available yet.
• The binomial theorem for a commutative ring. The proof given here uses only commutativity, associativity, distributivity and natural-number multiples, so the ring statement is true; it cannot be stated until rings exist (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides ↗).
• Group vocabulary for $\operatorname{Bij}(A)$. The count $n!$ is proved here, about a set of bijections. The group structure and the name symmetric group are The symmetric group $\operatorname{Sym}(X)$: the bijections of a set $X$ under composition ↗, later in the reading order, and a page there may cite the count from here.
• Asymptotics of $n!$, Stirling's formula among them. These need the logarithm and a good deal of integration, all far later in the reading order.

Every forward pointer above is orientation only: no item on this page depends on anything named in this section.