Combinatorial decompositions in the theory of algebraic enumeration

268 Pages·1979·13.49 MB·English

by  Ian Peter Goulden

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y •. 1
COMBINATORIAL DECOMPOSITIONS IN THE THEORY
. OP ALGEBRAIC’ ENUMERATION
i
; by .
*'
• k
Ian Peter Goulden
A thesis
presented to the University of Waterloo
^ n p a rtial fulfillm ent of the
requirements for the degree of
Doctor of Philosophy
in
S ta tistic s
VWaterloo, Ontario, 1979
(c) Ian Peter Goulden, 1979
I hereby declare that I am the.sole author 6f
this thesis. ' \
I authorize, the University of Waterloo to lend
this thesis to other institutions or individuals
for the purpose of scholarly research.
>
Signature
I further authorise the Universlty_jof Waterloo
to reproduce this thesis by photocopying or by
other means, in to ta l or in part, atAthe request
of other institutions or .individuals for the
purpose of scholarly research.
Signature
(11)
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Please sign below, and‘give address and date.
(iii)
ABSTRACT
Ther driving fprce behind- this thesis ^is-. ,th& d eli­
berate exploitation o f ‘the strong: conhecfion between
combinatorial structures, on the »one- ha|fd, and the "al-
tr * * Mh:‘Jfi.
.
. gebra of formal povfer .-series , on the ornier, The prin^ ' ■'
‘ ' " ' ' " M .. ‘ \
cipie which we adopt in the search foM general enqmera-
V - . ‘ . f -i
•> 7 ' f ii I'J ‘
tive theorems is that the basic compjlfiatorial-object is
the set and the basic enumerahive element is the gfener-
W '- y -
ating function. In applying this.; /pjrin-ciple, our object
.•i. . ■ M •
: is to obtain a combinatorial decomposition' of the set-*,
for which the set theoretic operations may be translated
straightforwardly into the a n aly tic operations on gener*'
ating functions Which are perm issible' for formal pbwer
' ’ ’ , . / ‘ " . • * •
series. To examine these general ideas, twd related
° ■*’ ' b V •
-
areas of combinatorial enumeration are considered, namely
a / ■ ■' i»
the enumeration of sequence^ and the enumeration o f-
‘ X-*’ , \ 'h 0 ,
rooted freb s. i; .
.'The enumeration ,of Sequences of in teg ers‘with pre­
scribed patterns of relationships between adjac^it ele­
ments is consideredl/rhis class contains many of the /
classical sequence and permutation problems. Th,e gener-
' ating functions* which, we employ are Eulerian for permu-
tations with'in-versions and ordinary for ‘ sequences, ajid .
i »
X • ' , ' - (iv) ■ „ .
a w ’ v -i 4 « ' .
are'obtained by-'a* number of decompositions 4of sequences* j
' ' ^ \ ’U \
r/™ * ,
into maximal, .tr,-paths and maximal • (%A;ir>,)-structures"; .
* 1 , "i O , Jp ■ * J. u.
' We 'demonstrate that, the requapqd generating-, functions '■*
“ ■ J; ' ' > a
,satisfy"b ‘system of linear equations in WHichtthe co-
■ i ■* * \ ’ » *
efficients are associated generating -functions i^iich
enumerate* ir^-paths and Milch may be obtained ;d ireh tly .
In mahy s tin c 'e h A speciarlizatlon_ to 'permutatibh. ^
' • •• • • • . . . .
g en erates fun.ctidps, is achieved by combinatorially d&T/
jfived transformations..of the'^^q'uehcel-gen'erat'in.gi function,
-The- extension,of ’these method's is given,.for “the enumera-
/ " » ia j1 . " * *' • «,. * <i
• s' ' . . ^ *■
tion of sequences->6f ' r-tuple®. of integers, ‘treated -els^-,
%Y- r , ». ” -y ‘ • „ ■ , ■
where’-by. binomial posets1; and fbr the* ^numeratipn 'p f,
* 4 • ‘ ’ « • i*' 1 ■* • •• *
■ . * * • 1 *\ ■ ' *'
.0 * S a . i . «. ■■ •**-
sequences- of- integers in ‘which -the prescribed'-patterfts s*v.,
* ‘ v ' ^ ' K 3 I ' ' i
involve relationships between pairs .of elements (which'
S •* ’ . . i ‘ ' >
pight■not be"adjacent.
In'the case of circufla*r sequence® witlj"respect to, \
• «» • 't ^ ^ * • .* 4 “
* '* K ‘ v ‘ * *■' • t.
.prescribed patte’rns', we deiftqQ'sira.te. that thdre i s a
striking relationship,-which-we c a ll’the "logarithmic
co'nnebtion", between the *,generating functions fdr the
linear sequence's previousiyv considered and the generating
functions fpr circular sequences. ‘ \
Entirely, different'm ethods’are used for the enumer-
* * ‘ ** « , 0
* > k
ation of sequences ‘with .respect to Jrfie' number of occur—
. ‘ ■ _' ■' ■' ’■ ■ i
renees as substrings in the sequence of each .element of
' , \
O' - >
a set of distinguished sequences. The decomposition .of
sequences in ternjs of clusters,' which are, sequences
formed by overlapping distinguished sequences, is com-
-bined with the Principle*of Inclusion and Exclusion to
° ‘ * Y A
, obtain the desired generating function.
s The enumeration of rooted treses is considered, in
which decompositions’ of rOotfed trees in terms of rooted
* ’
and. planted tipaes are used to (induce, systems, of func- ■
” * • I ■ . . . ’v\
tional equations for the.desired generating functions. ■
The combinatorial information which is retained across -
- these1 'decompositions includes vertex degree, colour and
adjacency information for properly K-coloured trees. . .
* ■ > ( ? » ! *
Solutions are explicitly determined'tty the use of a • •’ *
• special: form-of .the m ultivariate Lagrange theorem. A .
number of results is obtained, including the Matrix-Tree ‘
theorem and the de Bruijn-van Aardenrie Ehrenfest-Smith-
Tutte theorem, with emphasis placed on finding pairs o.fy
vsets whos^ elements are in one-one or .many^one .corres- '7
. " ■ \
porTdehce. • ' .
tvi)
ACKNOWLEDGEMENTS
The m aterial in this thesis jjs the' product of joint
work with Professor D.M. Jackson. The enthusiasm which
Professor'1 Jackson has for mathematics, the careful gui-
dan'oe which he has given me and the personal Interest
which’ h^ has taken in me have made my period of graduate
study an especially enriching one. I ain. also
grateful to him for .hi‘s advice and encouragement in the
preparation of this thesis. I would like to thank Mrs.
Sue. Embro for .'the great eare and sk ill which she has
. A
exercised^ in typing the manuscript, foils* work was com- ■
pleteci’under a scholarship from the Natural Sciences and
Engineering Research ‘Council of, Canada and I gratefully
acknowledge its, financial support.
i
(■Vi 17
- ' TABLE OF’CONTENTS
CHA.RjTER.1 -i Introduction and notation'1 '.‘1
^
# *
CHAPTER 2 - Enumerative methods for sequences with’ ‘ •
prescribed'pattern's 16
2.1 - Introduction and prelim inaries ’ • 16
2.2 - Inversions ’ <■ 21-
2.3 - The ordinary and Eulerian power series and
. sy's terns of^linear' equations v 23
\
2.4 - The rank-one update ’ ' > 30
4
2.5 - The encoding of combinatorial problems
.31
2.6 i TTj.'-pa^jMr'and specialisation to permutations ' - 34
2.7 - Generalization to the ^Iphabet Nr • 44
CHAPTER 3 - Applications to order one problems 50
4.
3.1 - Introduction and alternating sequences ■ » 50
3.2 - The Maximal Decomposition .theorem 56
3.3 - Distinguished terminal maximal paths 64
3.4 - D istinguished.initial and terminal maximal paths 69
3.5 - Terminal elements in maximal'paths 72
3.6 - The trip a rtite case 75
3.7 -j Inclusion and exclusion and trip a rtilio n s
/ with levels . . . 88
3.8 - Extension to r-arraj^i * .9 0
CHAPTER 4 --"Applications of arbitrary order- 93
' .
4.1 - Introduction add permutations with arbitrary
pattern ’ • ' - . " 93
4.2 - '(p,£)-periodic •permutations and a determinantal
identity ' 96
. - * .. • \
4.3 - (iT^,'h'2)-structures 106
4.4 - Extension to r-arrays 115
(viii)