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Generalizations of Rook Polynomials
A Dissertation
Presented to
The Faculty of the Graduate School of Arts and Sciences
Brandeis University
Department of Mathematics
Professor Ira M. Gessel, Advisor
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
Morris Dworkin
February, 1997
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This dissertation, directed and approved by the candidate’s
Committee, has been accepted and approved by the Graduate Faculty
of Brandeis University in partial fulfillment of the requirements for
the degree of
DOCTOR OF PHILOSOPHY
Dean of Arts and Sciences
Dissertation Committee^,
____
Ira \£3essel, Chair
f
SlCO (PaJcJL,
CUa
Susan Parker
S
Richard Stanley
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Thank you, Ira, for teaching me with such patience, generosity, and kindness.
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ABSTRACT
Generalizations of Rook Polynomials
A dissertation presented to the Faculty of
the Graduate School of Arts and Sciences of Brandeis
University, Waltham, Massachusetts
by Morris Dworkin
We study two generalizations of rook polynomials: Chung and Graham’s cover
polynomial, which keeps track of cycles in the digraph associated with the board,
and Gaxsia and Remmel’s 9-analogue.
In Part I we factor the cover polynomial completely for Ferrers boards with either
increasing or decreasing column heights. For column-permuted Ferrers boards, we
find a sufficient condition for partial factorization. We apply this result to several
special cases, including column-permuted “staircase boards,” getting a partial fac
torization in terms of the column permutation, as well as a sufficient condition for
complete factorization.
In Part II, for column-permuted Ferrers boards, we find a new recurrence for the
9-analogues of the hit numbers determined by Garsia and Remmel’s 9-rook numbers.
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We use the recurrence to define a statistic on permutations that constitutes a direct
combinatorial interpretation for these g-hit numbers. The statistic is Eulerian-
Mahonian on staircase boards. We use the statistic to prove a reciprocity theorem,
and we give a new proof that the sequence of coefficents in the q-hit numbers is
symmetric.
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CONTENTS
Part I: The Cover Polynomial
1 Introduction to Part I ...................................................................................1
2 Definitions......................................................................................................3
3 Factorization of the Cover Polynomial of Ferrers Boards..........................9
4 Partial Factorization of the Cover Polynomial of Skyline Boards . . . 19
5 Proof of the Partial Factorization Theorem...............................................27
6 Applications of the Partial Factorization Theorem....................................39
7 Partial Factorization of Column Permuted Staircase B oards.................43
Part II: g-Analogues
8 Introduction to Part II .............................................................................53
9 Permutations With Restricted Position...................................................55
10 Introduction to q-Analogues......................................................................57
11 Two g-Analogues of the Derivation of the Factorial Rook Polynomial . 59
12 Garsia and RemmePs g-Analogue..............................................................61
13 A New Recurrence for Garsia and Remmel’s g-Hit Num bers.................66
14 A Combinatorial Interpretation for the iV j...............................................71
15 A Reciprocity Theorem .............................................................................83
16 Symmetry Within the g-Hit Numbers.......................................................85
17 Other Questions.........................................................................................86
Appendix................................................................................................................88
R eferences.............................................................................................................90
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