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T. Amdeberhan
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Divisibility question while enumerating endomorphisms: PART 1

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T. Amdeberhan
  • 43.2k
  • 5
  • 57
  • 217

Divisibility question while enumerating endomorphisms

From endomorphisms of rank 1 of the full transformation semigroup $[n]^{[n]}$ or idempotents in $[n]^{[n]}$, we have $$c_n:=\sum_{m=1}^n\binom{n}mm^{n-m}.$$

QUESTION. Is it true that $n$ divides $c_n - 1$?