Three Representations for Set Partitions

For classes ${\cal O}$ of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order $\preceq$ (containment of permutations, subgraph relation etc.), we investigate restrictions on the function $f(n)$ counting objects with size $n$ in a lower ideal in $({\cal O},\preceq)$. We present a framework of edge $P$-colored complete graphs $({\cal C}(P),\preceq)$ which includes many of these situations, and we prove for it two such restrictions (jumps in growth): $f(n)$ is eventually constant or $f(n)\ge n$ for all $n\ge 1$; $f(n)\le n^c$ for all $n\ge 1$ for a constant $c>0$ or $f(n)\ge F_n$ for all $n\ge 1$, $F_n$ being the Fibonacci numbers. This generalizes a fragment of a more detailed theorem of Balogh, Bollobás and Morris on hereditary properties of ordered graphs.

Download Full-text

Counting visible levels in bargraphs and set partitions

Filomat ◽

10.2298/fil1919229c ◽

2019 ◽

Vol 33 (19) ◽

pp. 6229-6237 ◽

Cited By ~ 1

Author(s):

Nenad Cakic ◽

Toufik Mansour ◽

Rebecca Smith

Keyword(s):

Generating Functions ◽

Set Partitions

In this paper, we study the generating functions for the number of visible levels in compositions of n and set partitions of [n].

Download Full-text

Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

The Electronic Journal of Combinatorics ◽

10.37236/2550 ◽

2013 ◽

Vol 20 (1) ◽

Author(s):

Adam M Goyt ◽

Brady L Keller ◽

Jonathan E Rue

Keyword(s):

Fibonacci Numbers ◽

Natural Extension ◽

Statistical Distributions ◽

Set Partition ◽

Set Partitions ◽

Partition Statistics

We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most k. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles. We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.

Download Full-text