<s>
In	O
mathematics	O
,	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
–	I-Algorithm
Knuth	I-Algorithm
correspondence	I-Algorithm
,	O
also	O
referred	O
to	O
as	O
the	O
RSK	B-Algorithm
correspondence	I-Algorithm
or	O
RSK	B-Algorithm
algorithm	I-Algorithm
,	O
is	O
a	O
combinatorial	O
bijection	B-Algorithm
between	O
matrices	O
with	O
non-negative	O
integer	O
entries	O
and	O
pairs	O
of	O
semistandard	O
Young	O
tableaux	O
of	O
equal	O
shape	O
,	O
whose	O
size	O
equals	O
the	O
sum	O
of	O
the	O
entries	O
of	O
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
of	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
,	O
in	O
the	O
sense	O
that	O
taking	O
to	O
be	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
,	O
the	O
pair	O
will	O
be	O
the	O
pair	O
of	O
standard	O
tableaux	O
associated	O
to	O
the	O
permutation	B-Algorithm
under	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
.	O
</s>
<s>
The	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
–	I-Algorithm
Knuth	I-Algorithm
correspondence	I-Algorithm
extends	O
many	O
of	O
the	O
remarkable	O
properties	O
of	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
,	O
notably	O
its	O
symmetry	O
:	O
transposition	O
of	O
the	O
matrix	O
results	O
in	O
interchange	O
of	O
the	O
tableaux	O
.	O
</s>
<s>
The	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
is	O
a	O
bijective	B-Algorithm
mapping	I-Algorithm
between	O
permutations	B-Algorithm
and	O
pairs	O
of	O
standard	O
Young	O
tableaux	O
,	O
both	O
having	O
the	O
same	O
shape	O
.	O
</s>
<s>
This	O
bijection	B-Algorithm
can	O
be	O
constructed	O
using	O
an	O
algorithm	O
called	O
Schensted	O
insertion	O
,	O
starting	O
with	O
an	O
empty	O
tableau	O
and	O
successively	O
inserting	O
the	O
values	O
σ1	O
,,	O
σn	O
of	O
the	O
permutation	B-Algorithm
σ	O
at	O
the	O
numbers	O
1	O
,	O
2	O
,	O
n	O
;	O
these	O
form	O
the	O
second	O
line	O
when	O
σ	O
is	O
given	O
in	O
two-line	O
notation	O
:	O
</s>
<s>
The	O
definition	O
of	O
the	O
RSK	B-Algorithm
correspondence	I-Algorithm
reestablishes	O
symmetry	O
between	O
the	O
P	O
and	O
Q	O
tableaux	O
by	O
producing	O
a	O
semistandard	O
tableau	O
for	O
as	O
well	O
.	O
</s>
<s>
One	O
thus	O
obtains	O
a	O
bijection	B-Algorithm
from	O
matrices	O
to	O
ordered	O
pairs	O
,	O
of	O
semistandard	O
Young	O
tableaux	O
of	O
the	O
same	O
shape	O
,	O
in	O
which	O
the	O
set	O
of	O
entries	O
of	O
is	O
that	O
of	O
the	O
second	O
line	O
of	O
,	O
and	O
the	O
set	O
of	O
entries	O
of	O
is	O
that	O
of	O
the	O
first	O
line	O
of	O
.	O
</s>
<s>
The	O
above	O
definition	O
uses	O
the	O
Schensted	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
produces	O
a	O
standard	O
recording	O
tableau	O
,	O
and	O
modifies	O
it	O
to	O
take	O
into	O
account	O
the	O
first	O
line	O
of	O
the	O
two-line	O
array	O
and	O
produce	O
a	O
semistandard	O
recording	O
tableau	O
;	O
this	O
makes	O
the	O
relation	O
to	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
evident	O
.	O
</s>
<s>
It	O
is	O
natural	O
however	O
to	O
simplify	O
the	O
construction	O
by	O
modifying	O
the	O
shape	O
recording	O
part	O
of	O
the	O
algorithm	O
to	O
directly	O
take	O
into	O
account	O
the	O
first	O
line	O
of	O
the	O
two-line	O
array	O
;	O
it	O
is	O
in	O
this	O
form	O
that	O
the	O
algorithm	O
for	O
the	O
RSK	B-Algorithm
correspondence	I-Algorithm
is	O
usually	O
described	O
.	O
</s>
<s>
If	O
is	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
then	O
RSK	O
outputs	O
standard	O
Young	O
Tableaux	O
(	O
SYT	O
)	O
,	O
of	O
the	O
same	O
shape	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
are	O
SYT	O
having	O
the	O
same	O
shape	O
,	O
then	O
the	O
corresponding	O
matrix	O
is	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
As	O
a	O
result	O
of	O
this	O
property	O
by	O
simply	O
comparing	O
the	O
cardinalities	O
of	O
the	O
two	O
sets	O
on	O
the	O
two	O
sides	O
of	O
the	O
bijective	B-Algorithm
mapping	I-Algorithm
we	O
get	O
the	O
following	O
corollary	O
:	O
</s>
<s>
Suppose	O
the	O
RSK	B-Algorithm
algorithm	I-Algorithm
maps	O
to	O
then	O
the	O
RSK	B-Algorithm
algorithm	I-Algorithm
maps	O
to	O
,	O
where	O
is	O
the	O
transpose	O
of	O
.	O
</s>
<s>
In	O
particular	O
for	O
the	O
case	O
of	O
permutation	B-Algorithm
matrices	I-Algorithm
,	O
one	O
recovers	O
the	O
symmetry	O
of	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
:	O
</s>
<s>
Theorem	O
2	O
:	O
If	O
the	O
permutation	B-Algorithm
corresponds	O
to	O
a	O
triple	O
,	O
then	O
the	O
inverse	O
permutation	B-Algorithm
,	O
,	O
corresponds	O
to	O
.	O
</s>
<s>
This	O
leads	O
to	O
the	O
following	O
relation	O
between	O
the	O
number	O
of	O
involutions	O
on	O
with	O
the	O
number	O
of	O
tableaux	O
that	O
can	O
be	O
formed	O
from	O
(	O
An	O
involution	O
is	O
a	O
permutation	B-Algorithm
that	O
is	O
its	O
own	O
inverse	O
)	O
:	O
</s>
<s>
Conversely	O
,	O
if	O
is	O
any	O
permutation	B-Algorithm
corresponding	O
to	O
,	O
then	O
also	O
corresponds	O
to	O
;	O
hence	O
.	O
</s>
<s>
Let	O
and	O
let	O
the	O
RSK	B-Algorithm
algorithm	I-Algorithm
map	O
the	O
matrix	O
to	O
the	O
pair	O
,	O
where	O
is	O
an	O
SSYT	O
of	O
shape	O
.	O
</s>
<s>
Then	O
the	O
map	O
establishes	O
a	O
bijection	B-Algorithm
between	O
symmetric	O
matrices	O
with	O
row( )	O
and	O
SSYT	O
's	O
of	O
type	O
.	O
</s>
<s>
The	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
–	I-Algorithm
Knuth	I-Algorithm
correspondence	I-Algorithm
provides	O
a	O
direct	O
bijective	B-Algorithm
proof	O
of	O
the	O
following	O
celebrated	O
identity	O
for	O
symmetric	O
functions	O
:	O
</s>
