<s>
In	O
the	O
mathematics	O
of	O
permutations	B-Algorithm
and	O
the	O
study	O
of	O
shuffling	O
playing	O
cards	O
,	O
a	O
riffle	B-Algorithm
shuffle	I-Algorithm
permutation	I-Algorithm
is	O
one	O
of	O
the	O
permutations	B-Algorithm
of	O
a	O
set	O
of	O
items	O
that	O
can	O
be	O
obtained	O
by	O
a	O
single	O
riffle	O
shuffle	O
,	O
in	O
which	O
a	O
sorted	O
deck	O
of	O
cards	O
is	O
cut	O
into	O
two	O
packets	O
and	O
then	O
the	O
two	O
packets	O
are	O
interleaved	O
(	O
e.g.	O
</s>
<s>
Beginning	O
with	O
an	O
ordered	O
set	O
(	O
1	O
rising	O
sequence	O
)	O
,	O
mathematically	O
a	O
riffle	O
shuffle	O
is	O
defined	O
as	O
a	O
permutation	B-Algorithm
on	O
this	O
set	O
containing	O
1	O
or	O
2	O
rising	O
sequences	O
.	O
</s>
<s>
The	O
permutations	B-Algorithm
with	O
1	O
rising	O
sequence	O
are	O
the	O
identity	O
permutations	B-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
number	O
of	O
distinct	O
riffles	O
is	O
not	O
quite	O
the	O
sum	O
of	O
this	O
formula	O
over	O
all	O
choices	O
of	O
and	O
adding	O
to	O
(	O
which	O
would	O
be	O
)	O
,	O
because	O
the	O
identity	O
permutation	B-Algorithm
can	O
be	O
represented	O
in	O
multiple	O
ways	O
as	O
a	O
-shuffle	O
for	O
different	O
values	O
of	O
and	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
formula	O
for	O
this	O
number	O
is	O
;	O
for	O
instance	O
,	O
there	O
are	O
4503599627370444	O
riffle	B-Algorithm
shuffle	I-Algorithm
permutations	I-Algorithm
of	O
a	O
52-card	O
deck	O
.	O
</s>
<s>
In	O
this	O
model	O
,	O
the	O
identity	O
permutation	B-Algorithm
has	O
probability	O
of	O
being	O
generated	O
,	O
and	O
all	O
other	O
riffle	O
permutations	B-Algorithm
have	O
equal	O
probability	O
of	O
being	O
generated	O
.	O
</s>
<s>
A	O
pattern	B-Algorithm
in	O
a	O
permutation	B-Algorithm
is	O
a	O
smaller	O
permutation	B-Algorithm
formed	O
from	O
a	O
subsequence	O
of	O
some	O
values	O
in	O
the	O
permutation	B-Algorithm
by	O
reducing	O
these	O
values	O
to	O
the	O
range	O
from	O
1	O
to	O
while	O
preserving	O
their	O
order	O
.	O
</s>
<s>
Several	O
important	O
families	O
of	O
permutations	B-Algorithm
can	O
be	O
characterized	O
by	O
a	O
finite	O
set	O
of	O
forbidden	O
patterns	O
,	O
and	O
this	O
is	O
true	O
also	O
of	O
the	O
riffle	B-Algorithm
shuffle	I-Algorithm
permutations	I-Algorithm
:	O
they	O
are	O
exactly	O
the	O
permutations	B-Algorithm
that	O
do	O
not	O
have	O
321	O
,	O
2143	O
,	O
and	O
2413	O
as	O
patterns	O
.	O
</s>
<s>
Thus	O
,	O
for	O
instance	O
,	O
they	O
are	O
a	O
subclass	O
of	O
the	O
vexillary	B-Algorithm
permutations	I-Algorithm
,	O
which	O
have	O
2143	O
as	O
their	O
only	O
minimal	O
forbidden	O
pattern	B-Algorithm
.	O
</s>
<s>
When	O
a	O
deck	O
is	O
repeatedly	O
shuffled	O
using	O
these	O
permutations	B-Algorithm
,	O
it	O
remains	O
much	O
less	O
random	O
than	O
with	O
typical	O
riffle	O
shuffles	O
,	O
and	O
it	O
will	O
return	O
to	O
its	O
initial	O
state	O
after	O
only	O
a	O
small	O
number	O
of	O
perfect	O
shuffles	O
.	O
</s>
