---
title: "math-comb-recurrences — Mova code model"
description: "Mova code model of math/src/comb.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/math/src/comb.rs.groupant/recurrences.groupant/"
---
Elamur: [Mova](https://md.mova.elamur.ai/free/en/) · [Dictionary](https://md.dictionary.elamur.ai/free/en/) · [Research](https://md.research.elamur.ai/free/en/) · **Groupants**

This page as HTML: <https://groupants.elamur.ai/free/en/mova/math/src/comb.rs.groupant/recurrences.groupant/>

# math-comb-recurrences

The groupant beside math/src/comb.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

## Facts

- **kind:** groupant_fragment
- **status:** partial source bound model
- **source:** [`math/src/comb.rs`](https://github.com/ElamurAI/mova-free/blob/main/math/src/comb.rs)
- **source SHA-256:** `c5937a3576697c05…`
- **covers:** Stirling/Bell/partition/derangement/Fibonacci retain distinct zero seeds and update directions. Outputs have recurrence-specific nonunique index preimages. DP table/row costs differ; Cassini is not a complete sequence certificate. Child uses parent nt sieve/product joins. (bytes 4466–10809)
- **part of:** [math-comb](https://md.groupants.elamur.ai/free/en/mova/math/src/comb.rs.groupant/)
- **repository folder:** [`math/src/comb.rs.groupant/recurrences.groupant`](https://github.com/ElamurAI/mova-free/tree/main/math/src/comb.rs.groupant/recurrences.groupant)
- **repository commit:** `48f2e6f74146`

> **Implementation material.** A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.

## Components

| component | label | where |
|---|---|---|
| **approximant** | partial | [approximant](#approximant) |
| **ast** | partial | `../syntax.json` |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | partial | [invariant](#invariant) |
| **tests** | executed bounded checks | [tests](#tests) |

## Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

| id | type | label | where |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial source derived model | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial source derived model | [reverse_reconstruction](#reverse-reconstruction) |
| `stirling_state` | stirling_state | partial source derived model | [stirling_state](#stirling-state) |
| `partition_direction` | partition_direction | partial source derived model | [partition_direction](#partition-direction) |
| `empty_and_limit` | empty_and_limit | partial source derived model | [empty_and_limit](#empty-and-limit) |
| `doubling_cost` | doubling_cost | partial source derived model | [doubling_cost](#doubling-cost) |

## Model text


### descriptant

#### math-comb-recurrences

Stirling/Bell/partition/derangement/Fibonacci retain distinct zero seeds and update directions. Outputs have recurrence-specific nonunique index preimages. DP table/row costs differ; Cassini is not a complete sequence certificate. Child uses parent nt sieve/product joins.

Epoch 2026-10-31: complete source read, parent AST reused; partial finite evidence only.

### approximant

#### Scoped forward and cost

Stirling/Bell/partition/derangement/Fibonacci retain distinct zero seeds and update directions. Outputs have recurrence-specific nonunique index preimages. DP table/row costs differ; Cassini is not a complete sequence certificate. Child uses parent nt sieve/product joins.

No executable percent surrogate claimed.

### invariant

#### Scoped invariant

Stirling/Bell/partition/derangement/Fibonacci retain distinct zero seeds and update directions. Outputs have recurrence-specific nonunique index preimages. DP table/row costs differ; Cassini is not a complete sequence certificate. Child uses parent nt sieve/product joins.

Source range[4466,10809) bound by manifest digest. Parent domain applies, not arbitrary public states.

### models

## Source-scoped typed relations

### behavior_complement

stirling2(0,0) excludes0 and returns1; stirling2(n>3000,k>n) excludes zero shortcut because n limit wins. Empty Fibonacci seed excludes(1,1), returning(0,1).

### reverse_reconstruction

Partition count1 admits n=0 and n=1; derangement1 admits n=0 and n=2. Cassini acceptance constrains f1*(f1-f)-f²=(-1)^n but does not uniquely recover Fibonacci sequence provenance.

### stirling_state

S(i,j)=j*S(i-1,j)+S(i-1,j-1), descending j preserves previous row; reset row0=0 after each nonzero i. UnsignedI computes full row with multiplier i-1 and returns row sum.

### partition_direction

Distinct partitions update sum descending for each part; odd partitions update ascending. Euler pentagonal recurrence alternates pairs of signs at k(3k±1)/2. Their equality checks different counting paths.

### empty_and_limit

partitions0=1, partitions_k(0,0)=1, derangements0=1/1=0; Bell0=1. Each function has its own n gate; alternate Bell/partition/derangement checks cover smaller ranges.

### doubling_cost

F pair recursion halves n; c=a*(2b-a),d=a²+b² and parity chooses(c,d) or(d,c+d). O(log n) big arithmetic steps does not mean O(log n) bit time or bounded result storage.

### tests

#### Evidence

See ../tests.md for actual checks and source mutant; this fragment adds no runtime test.

Stirling/Bell/partition/derangement/Fibonacci retain distinct zero seeds and update directions. Outputs have recurrence-specific nonunique index preimages. DP table/row costs differ; Cassini is not a complete sequence certificate. Child uses parent nt sieve/product joins.