Open Access
2016 On singletons and adjacencies of set partitions
Augustine O. Munagi
Rocky Mountain J. Math. 46(1): 301-307 (2016). DOI: 10.1216/RMJ-2016-46-1-301


The number of singleton blocks in all partitions of a set $\{a_1,\ldots ,a_n\}$ is known to be equal to the number adjacencies, that is, pairs of consecutively numbered elements $(a_i,a_{i+1})$ in a block. We give a generalization of this relation by introducing the $d$-adjacency which is a pair of elements $(a_i,a_j)$ satisfying $j-i=d>0$. It is proved that the number of $d$-adjacencies in all partitions is independent of $d$. Then we show that the number of $d$-adjacencies in noncrossing partitions is a function of $d$ by means of an exact formula.


Download Citation

Augustine O. Munagi. "On singletons and adjacencies of set partitions." Rocky Mountain J. Math. 46 (1) 301 - 307, 2016.


Published: 2016
First available in Project Euclid: 23 May 2016

zbMATH: 1337.05008
MathSciNet: MR3506090
Digital Object Identifier: 10.1216/RMJ-2016-46-1-301

Primary: 05A15 , 05A18 , 05A19

Keywords: Adjacency , Catalan number , noncrossing partition , singleton

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 1 • 2016
Back to Top