6.5.4 Tensor Valence Slot Distribution

Tensor Valence Slot Distribution is the combinatorial framing of a valence sequence as a distribution of p contravariant markers among k total slot positions, equivalent to choosing which p of the k positions are up-arrows (with the remaining k - p automatically fixed as down-arrows), and covers the systematic enumeration of every such distribution, the identification of a canonical representative distribution for a given type, and the analogy between a valence sequence and a binary string. This framing treats valence not as an arbitrary sequence to be listed one instance at a time, but as an object governed by the same combinatorics as choosing a subset of positions from a fixed total.

Valence as a Choice of Positions

Reformulating Valence as Position Selection

Rather than listing a valence sequence symbol by symbol, it can be specified by giving the set S ⊆ {1, ..., k} of positions carrying an up-arrow, with |S| = p; the down-arrow positions are then simply the complement of S within {1, ..., k}, so the two descriptions — full sequence, or chosen subset — carry exactly the same information.

Counting Distributions via Binomial Coefficients

The number of ways to distribute p up-arrows among k positions is the binomial coefficient:

the same quantity already identified as the number of distinct valences for a fixed type (p, q) with q = k - p, now derived directly from the standard combinatorics of choosing a subset rather than from counting interleavings of two symbol types.

Valence as a Binary String

The Binary-String Analogy

Encoding an up-arrow as 1 and a down-arrow as 0 turns a valence sequence into a binary string of length k with exactly p ones, placing valence enumeration in direct correspondence with the standard combinatorial problem of listing all binary strings of a fixed length and fixed number of ones.

Diagram of All Distributions for k = 3, p = 2

Systematic Enumeration Order

The binary-string framing supports a standard systematic enumeration, such as listing all length-k strings with p ones in lexicographic order, giving a canonical, repeatable way to generate every valence compatible with a given type without omission or duplication.

The Canonical Sorted Distribution

Grouping All Upper Slots First

Among all C(k, p) distributions for a given type (p, q), one distribution is singled out by convention as canonical: the sorted one, with all p up-arrows placed first and all q down-arrows placed after, (↑, ..., ↑, ↓, ..., ↓), matching the standard way type is written in component notation, T^{i₁...i_p}_{j₁...j_q}.

Why a Canonical Representative Is Useful

When a discussion only needs to refer to "some tensor of type (p, q)" without caring about a specific arrangement, using the canonical sorted distribution avoids the need to specify or justify a choice among the C(k, p) otherwise equally valid distributions, simplifying notation whenever the specific arrangement is not itself the point under discussion.

Recovering an Arbitrary Distribution From the Canonical One

Any other distribution for the same type can be obtained from the canonical sorted distribution by a permutation of positions, so the canonical distribution together with a specific permutation is an alternative, fully equivalent way of specifying any valence, complementary to specifying the subset S directly.

Relation to the Other Valence Concepts

Slot Distribution Versus the Individual Counts

The contravariant count p and covariant count q are the two parameters fixing which binomial coefficient governs the distribution; the slot distribution itself is the specific choice made within the space that binomial coefficient counts, so distribution is a level of detail beyond the two counts and equal in detail to a full valence sequence.

Slot Distribution Versus the Ordered Pair

The ordered pair (p, q) identifies which distribution-counting problem is relevant (how many ways to place p ones among k slots); the slot distribution is one specific solution to that counting problem, corresponding to one specific valence out of the C(k,p) that the pair alone leaves undetermined.

Why the Distribution Framing Matters

Access to Standard Combinatorial Tools

Framing valence as a slot distribution makes the full toolkit of elementary combinatorics — binomial coefficients, subset enumeration, lexicographic ordering of binary strings — directly available for counting, generating, and comparing valences, rather than requiring bespoke reasoning about arrow sequences each time.

Justifying the Canonical Form Used Elsewhere

Recognizing the sorted distribution as one canonical choice among many combinatorially equivalent distributions explains, rather than merely asserts, why component notation conventionally writes every upper index before every lower index: it is simply the adoption of the canonical distribution as the default representative for a given type.