scholarly journals Permutation Statistics and $q$-Fibonacci Numbers

10.37236/190 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Adam M. Goyt ◽  
David Mathisen
Keyword(s):  
Fibonacci Numbers ◽  
Permutation Statistics ◽  
Set Partition ◽  
Set Partitions ◽  
Restricted Permutations ◽  
Mahonian Statistics ◽  
Partition Statistics

In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of $q$-Fibonacci numbers, which they related to $q$-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern restricted sets of permutations. We will give bijective proofs connecting some of our $q$-Fibonacci numbers to those of Carlitz, Cigler, Goyt and Sagan. We encode these permutations as words and use a weight to produce bijective proofs of $q$-Fibonacci identities. Finally, we study the distribution of some of these statistics on pattern restricted permutations that West showed were counted by even Fibonacci numbers.

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

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.

scholarly journals Set partition statistics and q-Fibonacci numbers

2009 ◽  
Vol 30 (1) ◽  
pp. 230-245 ◽  
Author(s):  
Adam M. Goyt ◽  
Bruce E. Sagan
Keyword(s):  
Fibonacci Numbers ◽  
Set Partition ◽  
Partition Statistics

scholarly journals On Growth Rates of Permutations, Set Partitions, Ordered Graphs and Other Objects

10.37236/799 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Martin Klazar
Keyword(s):  
Growth Rates ◽  
Fibonacci Numbers ◽  
Complete Graphs ◽  
Linear Orders ◽  
Set Partitions ◽  
Ordered Graphs ◽  
Containment Order ◽  
Hereditary Properties ◽  
Finite Linear ◽  
Lower Ideal

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.

Euler–Mahonian Statistics on Ordered Set Partitions

2008 ◽  
Vol 22 (3) ◽  
pp. 1105-1137 ◽  
Author(s):  
Masao Ishikawa ◽  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Ordered Set ◽  
Set Partitions ◽  
Ordered Set Partitions ◽  
Mahonian Statistics

scholarly journals Crossings and Nestings in Set Partitions of Classical Types

10.37236/392 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Martin Rubey ◽  
Christian Stump
Keyword(s):  
Type A ◽  
The Other ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Type D ◽  
Other Hand ◽  
The One

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.

scholarly journals Actions and Identities on Set Partitions

10.37236/1992 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Eric Marberg
Keyword(s):  
Abelian Group ◽  
Group Action ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Unitriangular Group

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $\mathbb{A}$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $\mathbb{A}^n$ on the set of $\mathbb{A}$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D.

scholarly journals A Generalization of Simion-Schmidt's Bijection for Restricted Permutations

10.37236/1686 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Astrid Reifegerste
Keyword(s):  
Symmetric Group ◽  
Permutation Statistics ◽  
Restricted Permutations ◽  
One To One ◽  
Special Case

We consider the two permutation statistics which count the distinct pairs obtained from the final two terms of occurrences of patterns $\tau_1\cdots\tau_{m-2}m(m-1)$ and $\tau_1\cdots\tau_{m-2}(m-1)m$ in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As a special case we derive a one-to-one correspondence between permutations which avoid each of the patterns $\tau_1\cdots\tau_{m-2}m(m-1)\in{\cal S}_m$ and those which avoid each of the patterns $\tau_1\cdots\tau_{m-2}(m-1)m\in{\cal S}_m$. For $m=3$ this correspondence coincides with the bijection given by Simion and Schmidt in [Europ. J. Combin. 6 (1985), 383-406].

scholarly journals Interpolating set partition statistics

1994 ◽  
Vol 68 (2) ◽  
pp. 262-295 ◽  
Author(s):  
Dennis White
Keyword(s):  
Set Partition ◽  
Partition Statistics

scholarly journals A Sperner-Type Theorem for Set-Partition Systems

10.37236/1987 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Karen Meagher ◽  
Lucia Moura ◽  
Brett Stevens
Keyword(s):  
Type Theorem ◽  
Higher Order ◽  
Set Partition ◽  
Set Partitions ◽  
Uniform Partition ◽  
Partition System ◽  
Sperner's Theorem

A Sperner partition system is a system of set partitions such that any two set partitions $P$ and $Q$ in the system have the property that for all classes $A$ of $P$ and all classes $B$ of $Q$, $A \not\subseteq B$ and $B \not\subseteq A$. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of Sperner's Theorem. In particular, we show that if $k$ divides $n$ the largest Sperner $k$-partition system on an $n$-set has cardinality ${n-1 \choose n/k-1}$ and is a uniform partition system. We give a bound on the cardinality of a Sperner $k$-partition system of an $n$-set for any $k$ and $n$.

scholarly journals Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

10.37236/8241 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Megan Ly
Keyword(s):  
Irreducible Representations ◽  
Character Theory ◽  
Set Partition ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Weyl Duality ◽  
Supercharacter Theory ◽  
Combinatorial Representation Theory ◽  
Centralizer Algebra

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field.  In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph. 

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