@MISC{CARDFIN2.MIZ,
  AUTHOR = {Kaliszyk, Cezary},
  TITLE = {Counting Derangements, Counting Non Bijective Functions and the
       Birthday Problem},
  NOTE = {This work has been partially supported by the KBN grant N519 385136},
  SECTION1 = {Preliminaries},
  SECTION2 = {Rounding},
  SECTION3 = {Counting Derangements},
  SECTION4 = {Counting Not-one-to-one Functions and the Birthday Problem},
  DAY = {23},
  MONTH = {February},
  YEAR = {2010},
  ADDRESS1 = {Institut f\"{u}r Informatik I4\\
              Techische Universit\"{a}t M\"{u}nchen\\
              Boltzmannstra{\ss}e 3\\
              85748 Garching, Germany},
  SUMMARY = {The article provides counting derangements of finite sets
  and counting non bijective functions. We provide a recursive formula
  for the number of derangements of a finite set, together with an
  explicit formula involving the number $e$. We count the number of
  non-one-to-one functions between to finite sets and perform a computation
  to give explicitely a formalization of the birthday problem in terms of
  probability. The article is an extension of \cite{CARD_FIN.ABS}.}}
