<s>
In	O
the	O
mathematical	O
field	O
of	O
combinatorics	O
,	O
jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
is	O
a	O
construction	O
due	O
to	O
which	O
defines	O
an	O
equivalence	O
relation	O
on	O
the	O
set	O
of	O
skew	O
standard	O
Young	O
tableaux	O
.	O
</s>
<s>
A	O
jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
slide	O
is	O
a	O
transformation	O
where	O
the	O
numbers	O
in	O
a	O
tableau	O
are	O
moved	O
around	O
in	O
a	O
way	O
similar	O
to	O
how	O
the	O
pieces	O
in	O
the	O
fifteen	O
puzzle	O
move	O
.	O
</s>
<s>
Two	O
tableaux	O
are	O
jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
equivalent	O
if	O
one	O
can	O
be	O
transformed	O
into	O
the	O
other	O
via	O
a	O
sequence	O
of	O
such	O
slides	O
.	O
</s>
<s>
"	O
Jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
"	O
(	O
literally	O
"	O
teasing	B-Algorithm
game	I-Algorithm
"	O
)	O
is	O
the	O
French	O
name	O
for	O
the	O
fifteen	O
puzzle	O
.	O
</s>
<s>
Jeu-de-taquin	O
slides	O
change	O
not	O
only	O
the	O
relative	O
order	O
of	O
the	O
entries	O
of	O
a	O
tableau	O
,	O
but	O
also	O
its	O
shape	O
.	O
</s>
<s>
In	O
the	O
definition	O
given	O
above	O
,	O
the	O
result	O
of	O
a	O
jeu-de-taquin	O
slide	O
is	O
given	O
as	O
a	O
skew	O
diagram	O
along	O
with	O
a	O
skew	O
standard	O
tableau	O
having	O
it	O
as	O
shape	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
it	O
is	O
useful	O
to	O
modify	O
the	O
above	O
definition	O
of	O
a	O
jeu-de-taquin	O
slide	O
in	O
such	O
a	O
way	O
that	O
,	O
when	O
given	O
a	O
skew	O
shape	O
along	O
with	O
a	O
skew	O
standard	O
tableau	O
and	O
an	O
addable	O
cell	O
as	O
an	O
input	O
,	O
it	O
yields	O
a	O
well-defined	O
skew	O
shape	O
along	O
with	O
a	O
skew	O
standard	O
tableau	O
at	O
its	O
output	O
.	O
</s>
<s>
Jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
slides	O
generalize	O
to	O
skew	O
semistandard	O
(	O
as	O
opposed	O
to	O
skew	O
standard	O
)	O
tableaux	O
and	O
retain	O
most	O
of	O
their	O
properties	O
in	O
that	O
generality	O
.	O
</s>
<s>
Two	O
skew	O
semistandard	O
tableaux	O
T	O
and	O
S	O
are	O
said	O
to	O
be	O
jeu-de-taquin	O
equivalent	O
if	O
one	O
can	O
transform	O
one	O
of	O
them	O
into	O
the	O
other	O
using	O
a	O
sequence	O
(	O
possibly	O
empty	O
)	O
of	O
slides	O
(	O
both	O
inward	O
and	O
outward	O
slides	O
being	O
allowed	O
)	O
.	O
</s>
<s>
Equivalently	O
,	O
two	O
skew	O
semistandard	O
tableaux	O
T	O
and	O
S	O
are	O
jeu-de-taquin	O
equivalent	O
if	O
and	O
only	O
if	O
they	O
have	O
the	O
same	O
rectification	O
.	O
</s>
<s>
It	O
can	O
then	O
be	O
shown	O
that	O
two	O
skew	O
semistandard	O
tableaux	O
T	O
and	O
S	O
are	O
jeu-de-taquin	O
equivalent	O
if	O
and	O
only	O
if	O
the	O
reading	O
words	O
of	O
T	O
and	O
S	O
are	O
Knuth	O
equivalent	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
the	O
rectification	O
of	O
a	O
skew	O
semistandard	O
tableau	O
T	O
can	O
also	O
be	O
obtained	O
as	O
the	O
insertion	O
tableau	O
of	O
the	O
reading	O
word	O
of	O
T	O
under	O
the	O
Robinson-Schensted	B-Algorithm
correspondence	I-Algorithm
.	O
</s>
<s>
Jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
can	O
be	O
used	O
to	O
define	O
an	O
operation	O
on	O
standard	O
Young	O
tableaux	O
of	O
any	O
given	O
shape	O
,	O
which	O
turns	O
out	O
to	O
be	O
an	O
involution	B-Algorithm
,	O
although	O
this	O
is	O
not	O
obvious	O
from	O
the	O
definition	O
.	O
</s>
<s>
Now	O
apply	O
a	O
jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
slide	O
to	O
turn	O
that	O
skew	O
tableau	O
into	O
a	O
straight	O
one	O
,	O
which	O
will	O
free	O
one	O
square	O
on	O
the	O
outside	O
border	O
.	O
</s>
<s>
Now	O
as	O
long	O
as	O
the	O
original	O
tableau	O
has	O
some	O
entries	O
left	O
,	O
repeat	O
the	O
operation	O
of	O
removing	O
the	O
entry	O
x	O
of	O
the	O
top-left	O
corner	O
,	O
performing	O
a	O
jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
slide	O
on	O
what	O
is	O
left	O
of	O
the	O
original	O
tableau	O
,	O
and	O
placing	O
the	O
value	O
−x	O
into	O
the	O
square	O
so	O
freed	O
.	O
</s>
<s>
Jeu	B-Algorithm
de	I-Algorithm
taquin	I-Algorithm
is	O
closely	O
connected	O
to	O
such	O
topics	O
as	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
–	I-Algorithm
Knuth	I-Algorithm
correspondence	I-Algorithm
,	O
the	O
Littlewood	O
–	O
Richardson	O
rule	O
,	O
and	O
Knuth	O
equivalence	O
.	O
</s>
