Why would a mathematician interested in combinatorics and cryptanalysis named lynne marie butler post the list below?

a lenient mulburryI'm about to tell you the answer, so delay reading on if you want to figure it out yourself. Each row of the list is a permutation of the multiset of letters in her name, and both combinatorics and cryptanalysis start with the study of permutations. The multiset that is being permuted has type (3,2,2,2,2,1,1,1,1,1,1,1) because each row has 3 e's, 2 l's, 2 n's, 2 r's, 2 blanks, and 1 each of the other 7 letters. (Curtis Greene can do this to your name. You can read more about elementary combinatorics in Brualdi's

tell riemann buyer

eat blurry linemen

eminent blurry ale

brainy menu teller

rumble realty nine

berlin realty menu

ninety mule barrel

mull binary entree

ninety marble rule

miller bunyan tree

eerily mental burn

An order-preserving surjection that relates subgroups of a finite
abelian p-group of type µ and subsets of a multiset of
type µ is illustrated by the
animation you'll see if you click
anywhere on the subgroup lattice shown above. This surjection is
defined in my paper *Order analogues and Betti polynomials*.
The animation that illustrates the case µ = (2,2,1) and p=2
was produced with the help of Toby Orloff, using software
developed by the
Geometry Center. It suggests
that if [µ,k] is the number of subgroups of order p^{k} in a finite abelian p-group of type µ, then

[(2,2,1),5] = 1because there is 1 subgroup at the top level 5, 1+2+4 subgroups at level 4, 1+2+2(4)+8 subgroups at levels 3 and 2, 1+2+4 subgroups at level 1, and 1 subgroup at the bottom level 0. In my first paper, A unimodality result in the enumeration of subgroups of a finite abelian group, I proved that [µ,k]≤[µ,k+1] if k < n/2 and µ is a partition of n. The subgroup lattice doesn't narrow as you move toward its middle level!

[(2,2,1),4] = 1 + p + p^{2}

[(2,2,1),3] = 1 + p + 2 p^{2}+ p^{3}

[(2,2,1),2] = 1 + p + 2 p^{2}+ p^{3}

[(2,2,1),1] = 1 + p + p^{2}

[(2,2,1),0] = 1

The h-vector of a simplicial polytope is computed from its f-vector, which counts the number of faces in each dimension. So, for example, the f-vector of an octahedron is (6,12,8) because it has 6 vertices, 12 edges and 8 triangular facets. The h-vector of an octahedron is (1,3,3,1).

For type (1,1,...,1), this result is the "p-analogue" of the fact that the number of subsets of cardinality k equals the number of subsets of cardinality n-k in a set of cardinality n. A combinatorial proof of this fact pairs each subset with its complement.

The regular three-dimensional polytopes are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron; three of these five Platonic solids have triangular facets. There are six regular four-dimensional polytopes; the one that has octahedral facets is called the 24-cell.

The Möbius function for partially ordered sets generalizes the Möbius function µ in number theory. If n is a product of k distinct primes, then µ(n) is 1 if k is even and -1 if k is odd. InEnumerative Combinatorics, Richard Stanley explains the topological significance of the Möbius function for partially ordered sets.

** Laurie Rubel*** Noncooperative Zero and General Sum Games*, senior thesis, Haverford College, 1992.
Rubel's thesis was inspired by Butler's unpublished work on game-theoretic analyses of matching algorithms.
This work, in turn, was initiated to study the *auction algorithm* that Butler invented and Maley implemented
for Princeton University to place students into limited enrollment courses. Rubel went on to earn a masters degree at
Tel Aviv University Graduate School of Education. In 2002 she earned a Ph.D. in Research in Mathematics Education from
Teachers College of Columbia University. Her dissertation, entitled "Probabilistic Misconceptions: Middle and High School Students' Judgments
Under Uncertainty" was guided by Henry Pollak. She is now a postdoc at the University of Wisconsin at Madison.

** Eric Muhlheim*** Some computations on subgroup lattice
embeddings*, senior thesis, Princeton University, 1991. The computations in this thesis are acknowledged in *Generalized flags in p-groups*, a paper coauthored by Butler and Hales. Muhlheim went on to work for Morgan Stanley in New York.

**Michael Aguilar*** Combinatorial properties of the lattice of subgroups of a finite abelian p-group*, senior thesis, Princeton University, 1990. Aguilar proved that
[µ,k]^{2}≥[µ,k-1][µ,k+1] for µ=(2,1,...,1), a special case of Butler's conjecture in *The q-log-concavity of q-binomial coefficients* that the lattice of subgroups of any finite abelian group is rank-log-concave. Aguilar went on to the Ph.D. program in mathematics at The University of Chicago supported by an NSF Graduate Fellowship for Minorities.

Return to the home page of Lynne Butler.

Return to the home page of the
Department of Mathematics at
Haverford College.

This page was created by lbutler@haverford.edu. It was last updated 3/27/03.