### Abstract:

In this thesis we consider the problem of recognizing and generating permutations
of the integers
f
0
;:::;
2
r
°
1
g
that have a special property. The idea is as follows.
Any integer from
f
0
;:::;
2
r
°
1
g
can be represented in its binary form as a bit string
of length
r
. Consequently, a permutation
p
= (
p
0
;p
1
;:::;p
2
r
°
1
)
of
f
0
;:::;
2
r
°
1
g
can be represented as a bit string, call it
R
(
p
)
, of length
r:
2
r
by concatenating the
binary representations of integers in the permutation. Imagine now the bit string
R
(
p
)
to be arranged circularly. Our question then is, to obtain a permutation of
f
0
;:::;
2
r
°
1
g
, can we start reading of binary representations of length
r
, from any bit on this
circle? Clearly, by our definition of
R
(
p
)
there is at least one such valid start bit. The
permutation
p
is said to be stable if any bit of
R
(
p
)
(arranged circularly) could serve as
the start bit for finding a permutation of
f
0
;:::;
2
r
°
1
g
.
In this thesis we prove several interesting properties of stable permutation. Algo-
rithm for recognizing stable permutations, rules to generate stable permutations are also
proposed.