<s>
In	O
the	O
mathematical	O
study	O
of	O
permutations	B-Algorithm
and	O
permutation	B-Algorithm
patterns	I-Algorithm
,	O
a	O
superpattern	B-Algorithm
or	O
universal	O
permutation	B-Algorithm
is	O
a	O
permutation	B-Algorithm
that	O
contains	O
all	O
of	O
the	O
patterns	O
of	O
a	O
given	O
length	O
.	O
</s>
<s>
More	O
specifically	O
,	O
a	O
k-superpattern	O
contains	O
all	O
possible	O
patterns	O
of	O
length	O
k	O
.	O
</s>
<s>
If	O
π	O
is	O
a	O
permutation	B-Algorithm
of	O
length	O
n	O
,	O
represented	O
as	O
a	O
sequence	O
of	O
the	O
numbers	O
from	O
1	O
to	O
n	O
in	O
some	O
order	O
,	O
and	O
s=	O
s1	O
,	O
s2	O
,	O
...	O
,	O
sk	O
is	O
a	O
subsequence	O
of	O
π	O
of	O
length	O
k	O
,	O
then	O
s	O
corresponds	O
to	O
a	O
unique	O
pattern	O
,	O
a	O
permutation	B-Algorithm
of	O
length	O
k	O
whose	O
elements	O
are	O
in	O
the	O
same	O
order	O
as	O
s	O
.	O
That	O
is	O
,	O
for	O
each	O
pair	O
i	O
and	O
j	O
of	O
indexes	O
,	O
the	O
ith	O
element	O
of	O
the	O
pattern	O
for	O
s	O
should	O
be	O
less	O
than	O
the	O
jthe	O
element	O
if	O
and	O
only	O
if	O
the	O
ith	O
element	O
of	O
s	O
is	O
less	O
than	O
the	O
jth	O
element	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
π	O
is	O
the	O
permutation	B-Algorithm
25314	O
,	O
then	O
it	O
has	O
ten	O
subsequences	O
of	O
length	O
three	O
,	O
forming	O
the	O
following	O
patterns	O
:	O
</s>
<s>
A	O
permutation	B-Algorithm
π	O
is	O
called	O
a	O
k-superpattern	O
if	O
its	O
patterns	O
of	O
length	O
k	O
include	O
all	O
of	O
the	O
length-k	O
permutations	B-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
length-3	O
patterns	O
of	O
25314	O
include	O
all	O
six	O
of	O
the	O
length-3	O
permutations	B-Algorithm
,	O
so	O
25314	O
is	O
a	O
3-superpattern	O
.	O
</s>
<s>
No	O
3-superpattern	O
can	O
be	O
shorter	O
,	O
because	O
any	O
two	O
subsequences	O
that	O
form	O
the	O
two	O
patterns	O
123	O
and	O
321	O
can	O
only	O
intersect	O
in	O
a	O
single	O
position	O
,	O
so	O
five	O
symbols	O
are	O
required	O
just	O
to	O
cover	O
these	O
two	O
patterns	O
.	O
</s>
<s>
introduced	O
the	O
problem	O
of	O
determining	O
the	O
length	O
of	O
the	O
shortest	O
possible	O
k-superpattern	O
.	O
</s>
<s>
He	O
observed	O
that	O
there	O
exists	O
a	O
superpattern	B-Algorithm
of	O
length	O
k2	O
(	O
given	O
by	O
the	O
lexicographic	O
ordering	O
on	O
the	O
coordinate	O
vectors	O
of	O
points	O
in	O
a	O
square	O
grid	O
)	O
and	O
also	O
observed	O
that	O
,	O
for	O
a	O
superpattern	B-Algorithm
of	O
length	O
n	O
,	O
it	O
must	O
be	O
the	O
case	O
that	O
it	O
has	O
at	O
least	O
as	O
many	O
subsequences	O
as	O
there	O
are	O
patterns	O
.	O
</s>
<s>
The	O
upper	O
bound	O
of	O
k2	O
on	O
superpattern	B-Algorithm
length	O
proven	O
by	O
Arratia	O
is	O
not	O
tight	O
.	O
</s>
<s>
proved	O
that	O
there	O
is	O
a	O
k-superpattern	O
of	O
length	O
at	O
most	O
k( k+1	O
)	O
/2	O
for	O
every	O
k	O
.	O
</s>
<s>
conjectured	O
that	O
the	O
true	O
length	O
of	O
the	O
shortest	O
k-superpattern	O
is	O
asymptotic	O
to	O
k2/2	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
in	O
contradiction	O
with	O
a	O
conjecture	O
of	O
Alon	O
on	O
random	O
superpatterns	B-Algorithm
described	O
below	O
.	O
</s>
<s>
Researchers	O
have	O
also	O
studied	O
the	O
length	O
needed	O
for	O
a	O
sequence	O
generated	O
by	O
a	O
random	O
process	O
to	O
become	O
a	O
superpattern	B-Algorithm
.	O
</s>
<s>
observes	O
that	O
,	O
because	O
the	O
longest	O
increasing	O
subsequence	O
of	O
a	O
random	O
permutation	B-Algorithm
has	O
length	O
(	O
with	O
high	O
probability	O
)	O
approximately	O
2√n	O
,	O
it	O
follows	O
that	O
a	O
random	O
permutation	B-Algorithm
must	O
have	O
length	O
at	O
least	O
k2/4	O
to	O
have	O
high	O
probability	O
of	O
being	O
a	O
k-superpattern	O
:	O
permutations	B-Algorithm
shorter	O
than	O
this	O
will	O
likely	O
not	O
contain	O
the	O
identity	O
pattern	O
.	O
</s>
<s>
He	O
attributes	O
to	O
Alon	O
the	O
conjecture	O
that	O
,	O
for	O
any	O
,	O
with	O
high	O
probability	O
,	O
random	O
permutations	B-Algorithm
of	O
length	O
will	O
be	O
k-superpatterns	O
.	O
</s>
