Bell numbers

Concept

Bell numbers are the numbers of set partitions: the nth Bell number b_n is the number of partitions of an n-element set. In the provided sources, they are given with the recurrence b_0 = 1 and b_{n+1} = \sum_{k=0}^n \binom{n}{k} b_k, and they are used to count possible index-relationship patterns in array-oriented test generation and to state hard bounds in hashing/min-entropy condenser analysis.

First seen 5/31/2026

Last seen 6/5/2026

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Definition

Bell numbers count set partitions. The provided TACAS source states that the number of partitions of an n-set is the nth Bell number, written (b_n).

Recurrence

The same source gives the following recurrence:

(b_0 = 1)
(b_{n+1} = \sum_{k=0}^{n} \binom{n}{k} b_k), for (n \ge 0)
with (\binom{n}{k} = \frac{n!}{k!(n-k)!})

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NEIGHBORHOOD

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RELATIONSHIPS

3 connections

Code-based Test Generation for Validation of Functional Processor Descriptions ← uses 90% 2e

The paper uses Bell numbers to characterize the combinatorial complexity of index relationships.

Code-based Test Generation for Validation of Functional Processor Descriptions ← mentions 100% 1e

The paper mentions Bell numbers in the context of array index relationship enumeration.

array constraints ← uses 87% 1e

The number of index relationships in array constraints corresponds to Bell numbers

LINKED ENTITIES

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Code-based Test Generation for Validation of Functional Processor Descriptions USES Extracted graph relationship

array constraints USES Extracted graph relationship

CITATIONS

5 sources

5 citations — click to expand

[1] Bell numbers count partitions of an n-element set. Fabrice.Baray,Henri.Michel

[2] The provided source gives the recurrence b_0 = 1 and b_{n+1} = sum_{k=0}^n binom(n,k) b_k, with binom(n,k) = n!/(k!(n-k)!). Fabrice.Baray,Henri.Michel

[3] In array-oriented test generation, Bell numbers are used to count possible relationship patterns among indices, motivating an alias-coverage style criterion. Fabrice.Baray,Henri.Michel

[4] The TACAS source describes the Bell-number-based enumeration as having exponential complexity, while noting that this is not necessarily problematic in its search setting. Fabrice.Baray,Henri.Michel

[5] The arXiv paper on q-wise independence tails states that it provides hard bounds in terms of Bell numbers. Lower bounds on q-wise independence tails and applications to min-entropy condensers