Math 2.0

Combinatorics Theorem

Formula for the number of surjections between finite sets

Builds on Surjective functionBinomial coefficient

Statement

Let S(n,m)S(n,m) be the number of surjective functions from the set [n]={1,…,n}[n] = \{1, \dots, n\} onto the set [m]={1,…,m}[m] = \{1, \dots, m\}. Then S(n,m)=∑k=0m(−1)k(mk)(m−k)n,S(n,m) = \sum_{k=0}^{m} (-1)^k \binom{m}{k} (m-k)^n, where (mk)\binom{m}{k} is a binomial coefficient.

Remarks

The term (m−k)n(m-k)^n is the number of all functions from [n][n] to a fixed set of m−km-k elements. The sum comes from the inclusion-exclusion principle: count all functions, then correct for the functions that miss some chosen set of kk values of [m][m].

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Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_inclusion-exclusion_surjections.ptx#L29, source/s_inclusion-exclusion_surjections.ptx:29
About this page
TagX9JC3BH2T0
URIm2://X9JC3BH2T0
Kindstatement, theorem
WritingW1 stated
MSC05A05, 05A19
StatusThis copy of the site has no build data. Run m2 status X9JC3BH2T0.
UsesSurjective function, Binomial coefficient
Changed4 November 2026

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