<s>
In	O
combinatorial	O
mathematics	O
,	O
a	O
Stirling	B-Algorithm
permutation	I-Algorithm
of	O
order	O
k	O
is	O
a	O
permutation	B-Algorithm
of	O
the	O
multiset	B-Language
1	O
,	O
1	O
,	O
2	O
,	O
2	O
,	O
...	O
,	O
k	O
,	O
k	O
(	O
with	O
two	O
copies	O
of	O
each	O
value	O
from	O
1	O
to	O
k	O
)	O
with	O
the	O
additional	O
property	O
that	O
,	O
for	O
each	O
value	O
i	O
appearing	O
in	O
the	O
permutation	B-Algorithm
,	O
the	O
values	O
between	O
the	O
two	O
copies	O
of	O
i	O
are	O
larger	O
than	O
i	O
.	O
</s>
<s>
The	O
number	O
of	O
Stirling	B-Algorithm
permutations	I-Algorithm
of	O
order	O
k	O
is	O
given	O
by	O
the	O
double	O
factorial	O
(	O
2k1	O
)	O
!!	O
.	O
</s>
<s>
Stirling	B-Algorithm
permutations	I-Algorithm
were	O
introduced	O
by	O
in	O
order	O
to	O
show	O
that	O
certain	O
numbers	O
(	O
the	O
numbers	O
of	O
Stirling	B-Algorithm
permutations	I-Algorithm
with	O
a	O
fixed	O
number	O
of	O
descents	O
)	O
are	O
non-negative	O
.	O
</s>
<s>
Stirling	B-Algorithm
permutations	I-Algorithm
may	O
be	O
used	O
to	O
describe	O
the	O
sequences	O
by	O
which	O
it	O
is	O
possible	O
to	O
construct	O
a	O
rooted	O
plane	O
tree	O
with	O
k	O
edges	O
by	O
adding	O
leaves	O
one	O
by	O
one	O
to	O
the	O
tree	O
.	O
</s>
<s>
For	O
,	O
if	O
the	O
edges	O
are	O
numbered	O
by	O
the	O
order	O
in	O
which	O
they	O
were	O
inserted	O
,	O
then	O
the	O
sequence	O
of	O
numbers	O
in	O
an	O
Euler	B-Operating_System
tour	I-Operating_System
of	O
the	O
tree	O
(	O
formed	O
by	O
doubling	O
the	O
edges	O
of	O
the	O
tree	O
and	O
traversing	O
the	O
children	O
of	O
each	O
node	O
in	O
left	O
to	O
right	O
order	O
)	O
is	O
a	O
Stirling	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
Conversely	O
every	O
Stirling	B-Algorithm
permutation	I-Algorithm
describes	O
a	O
tree	O
construction	O
sequence	O
,	O
in	O
which	O
the	O
next	O
edge	O
closer	O
to	O
the	O
root	O
from	O
an	O
edge	O
labeled	O
i	O
is	O
the	O
one	O
whose	O
pair	O
of	O
values	O
most	O
closely	O
surrounds	O
the	O
pair	O
of	O
i	O
values	O
in	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
Stirling	B-Algorithm
permutations	I-Algorithm
have	O
been	O
generalized	O
to	O
the	O
permutations	B-Algorithm
of	O
a	O
multiset	B-Language
with	O
more	O
than	O
two	O
copies	O
of	O
each	O
value	O
.	O
</s>
<s>
Researchers	O
have	O
also	O
studied	O
the	O
number	O
of	O
Stirling	B-Algorithm
permutations	I-Algorithm
that	O
avoid	O
certain	O
patterns	O
.	O
</s>
