The algebra
F
n
{\mathcal {F}_n}
generated by
E
1
,
…
,
E
n
−
1
{E_1},\; \ldots \;,\;{E_{n - 1}}
subject to the defining relations
E
i
2
=
x
i
E
i
(
i
=
1
,
…
,
n
−
1
)
,
E
i
+
1
E
i
E
i
+
1
=
y
i
E
i
+
1
(
i
=
1
,
…
,
n
−
2
)
,
E
i
E
j
=
E
j
E
i
(
|
i
−
j
|
⩾
2
)
E_i^2 = {x_i}{E_i}\;(i = 1,\; \ldots \;,\;n - 1),\;{E_{i + 1}}{E_i}{E_{i + 1}} = {y_i}{E_{i + 1}}\;(i = 1,\; \ldots \;,\;n - 2),\;{E_i}{E_j} = {E_j}{E_i}\;(|i - j| \geqslant 2)
is shown to be a semisimple algebra of dimension
n
!
n!
if the parameters
x
1
,
…
,
x
n
−
1
,
y
1
,
…
,
y
n
−
2
{x_1},\; \ldots \;,\;{x_{n - 1}},\;{y_1},\; \ldots \;,\;{y_{n - 2}}
are generic. We also prove that the Bratteli diagram of the tower
(
F
n
)
n
⩾
0
{({\mathcal {F}_n})_{n \geqslant 0}}
of these algebras is the Hasse diagram of the Young-Fibonacci lattice, which is an interesting example, as well as Young’s lattice, of a differential poset introduced by
R
\operatorname {R}
. Stanley. A Young-Fibonacci analogue of the ring of symmetric functions is given and studied.