Transcript Enumerating kth Roots in the Symmetric Inverse Monoid

Enumerating kth Roots in the
Symmetric Inverse Monoid
Christopher W. York
Dr. Valentin V. Andreev, Mentor
Department of Mathematics
October 1, 2014
The Symmetric Inverse Monoid
• Denoted SIM(n), the symmetric inverse monoid is the set of all
partial one-to-one mappings from the set {1,2,…,n} onto itself with
the operation of composition
1 2 3 4 5
• For example, 𝜎 =
∈ 𝑆𝐼𝑀(5) maps 1 to 5, 2 to
5 2 1 3 −
itself, 3 to 1, 4 to 3, and 5 to nothing.
Cycle and Path Notation
• Every element in SIM(n) can be expressed as the product of disjoint
paths and cycles
• Paths map a number to the one next to it and the last number to
nothing and are denoted with brackets. For example, [12357] maps 1
to 2, 2 to 3, 3 to 5, 5 to 7, and 7 to nothing
• Cycles map the last number to the first number and are denoted with
parenthesis. For example, (3452) maps 3 to 4, 4 to 5, 5 to 2, and 2 to 3
• Length of a path or cycle is the number of numbers in it. For example,
[12357] is of length 5.
Raising Elements to a Power k
• Raising an element in SIM(n) to the 𝑘th power means applying the
mapping unto itself 𝑘 times, creating a “skipping by 𝑘” effect
• For example, 123456789 4 = 159 26 37 [48]
• This “breaks” a path into 𝑘 paths of lengths differing by at most 1
• Fact: Let 𝜎 = 𝜎1 𝜎2 … 𝜎𝑚 ∈ 𝑆𝐼𝑀(𝑛) where 𝜎1,2,…,𝑚 are disjoint paths
𝑘
and/or cycles. Then 𝜎 𝑘 = 𝜎1𝑘 𝜎2𝑘 … 𝜎𝑚
Definition of kth Root
• An element 𝛼 ∈ 𝑆𝐼𝑀(𝑛) is a kth root of 𝜎 if and only if 𝛼 𝑘 = 𝜎.
• For example, in 123456789 4 = 159 26 37 [48], [123456789] is
a 4th root of [159][26][37][48].
• The aim is to find formulas to determine the number of kth roots
any element of SIM(n).
Previous Research
• Annin et al. [2] first determined whether an element in the symmetric
group, an algebraic structure similar to SIM(n), has a kth root
• Recently, Annin [1] determined whether an element in SIM(n) has a
kth root
• both papers posed the question of how many kth roots an element has
Interlacing Paths
• Raising a path to the 𝑘th power breaks it into 𝑘 paths, so creating a 𝑘th
root of an element would be the “interlacing” of paths in groups of 𝑘.
• If all the paths can’t be legally interlaced, then there are no 𝑘th roots
of the element
• Ex.: The interlacing of [123], [45], [67], and [89] would be
[146825793]. Clearly, 146825793 4 = 123 45 67 [89].
• The order of paths in the interlacing matters
• There can only be 𝑘 paths starting with the longest paths and lengths
varying by at most 1.
The Root Counting Function
• The number of distinct 𝑘th roots an element 𝜎 ∈ 𝑆𝐼𝑀(𝑛) will be
denoted by 𝜑(𝜎, 𝑘).
• This is equivalent to the number of ways to interlace all the paths of 𝜎
in groups of 𝑘.
A Simple Case
• Let 𝜎 = 𝜎1 𝜎2 … 𝜎𝑟𝑘 ∈ 𝑆𝐼𝑀 𝑛 where 𝜎1,2,…,𝑟𝑘 are disjoint paths of the
same length all greater than 1.
• Then 𝜑 𝜎, 𝑘 =
𝑟𝑘 !
.
𝑟!
• There are 𝑟 interlacings, whose order doesn’t matter
• The order within the interlacings does matter
A slightly More Complex Case
• Let 𝜎 = 𝜎1 𝜎2 … 𝜎𝑟𝑘 where 𝜎1,2,…,𝑟𝑘 are disjoint paths and the lengths
of 𝜎1 , 𝜎2 , … , 𝜎𝑟𝑘−1 are equal and 𝜎𝑟𝑘 is of length 1 less than the others.
All lengths are greater than 1.
• Then 𝜑 𝜎, 𝑘 =
1 𝑟𝑘 !
𝑘 𝑟!
=
𝑟𝑘−1 !
.
𝑟−1 !
• Again, there are 𝑟 interlacings, whose order doesn’t matter
• The smaller path has to be at the end of the interlacing it’s in
• There is a 1/𝑘 probability that the smaller path will be in a right place
An element with two weakly varying lengths
• Let 𝜎 be the product of 𝑟𝑘 disjoint paths where the first 𝑟𝑘 − 𝑚 paths
are equal length and the other paths are of length 1 less those paths.
All lengths are greater than 1.
• The general form of the number of roots is
𝑟𝑘 !
𝜑 𝜎, 𝑘 = 𝑃𝑟𝑜𝑏(𝑟𝑖𝑔ℎ𝑡 𝑝𝑙𝑎𝑐𝑒)
𝑟!
• 𝑃𝑟𝑜𝑏 𝑟𝑖𝑔ℎ𝑡 𝑝𝑙𝑎𝑐𝑒 is the probability the smaller paths will be in the
right places in the interlacings
Paths of length 1
• Fact: 𝑎1 𝑎2 … 𝑎𝑛 𝑘 = 𝑎1 𝑎2 … [𝑎𝑛 ] whenever 𝑛 ≤ 𝑘.
• This is the only instance where raising a path to the 𝑘th power
“breaks” it into less than 𝑘 paths
• Therefore, proper interlacings of paths only length 1 can have any
number of paths as long as it’s at most 𝑘 paths.
• For example, if 𝜎 = 1 2 3 [4] and 𝑘 = 4, proper interlacings of 𝜎’s
paths can include [12], [1], [134], [1234].
• Partitions of the number of paths can represent this
Some Helpful Formulas
• Let 𝜎 = 𝛼𝛽 where 𝛼 is a product of disjoint paths and 𝛽 is a product
of disjoint cycles
• Then 𝜑 𝜎, 𝑘 = 𝜑 𝛼, 𝑘 𝜑(𝛽, 𝑘).
• Paths and cycles cannot be interlaced
• Let 𝜎 = 𝛼𝛽 where 𝛼 and 𝛽 are products of disjoint paths such that all
paths in 𝛼 are at longer than those in 𝛽 by at least 2.
• Then 𝜑 𝜎, 𝑘 = 𝜑 𝛼, 𝑘 𝜑(𝛽, 𝑘).
• Paths varying by lengths of more than one cannot be interlaced
Further Research
• Elements with more than two varying lengths
• Elements with cycles
• Elements with weakly varying lengths starting with paths length 1
• Creating programs to calculate the number roots
• Thank you for listening!
References
[1] Annin, S. et al., On k’th roots in the symmetric inverse monoid. Pi
Mu Epsilon 13:6 (2012), 321-331.
[2] Annin, S., Jansen, T. and Smith, C., On k’th roots in the symmetric
and alternating Groups, Pi Mu Epsilon Journal 12:10 (2009), 581-589.