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Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions
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Mark Shattuck
Published/Copyright:
May 31, 2010
Abstract
A finite set partition is said to have a descent at i if it has a descent at i in its canonical representation as a restricted growth function (and likewise for level and rise). In this note, we provide direct combinatorial proofs as well as extensions of recent formulas for the total number of rises, levels, and descents in all the partitions of an n-set with a prescribed number of blocks. In addition, we supply direct proofs of formulas for the number of partitions having a fixed number of levels.
Received: 2009-10-12
Accepted: 2009-12-17
Published Online: 2010-05-31
Published in Print: 2010-May
© de Gruyter 2010
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Articles in the same Issue
- Recounting the Number of Rises, Levels, and Descents in Finite Set Partitions
- Multiplicities of Integer Arrays
- Adjugates of Diophantine Quadruples
- A Binary Tree Representation for the 2-Adic Valuation of a Sequence Arising From a Rational Integral
- On a Variant of Van Der Waerden's Theorem
- Note on a Result of Haddad and Helou
- Sequences of Density ζ(k) – 1
- Square-Full Divisors of Square-Full Integers
- Lower Bounds for the Principal Genus of Definite Binary Quadratic Forms
