re crank form expressions as they are, by induction on the lengths of the words in the alphabets ${\cal L}_n$, any word turn out to be equal to (a scalar multiple) of the standard expression of a seat-plan $w$ of $\Sigma_n^1$. Hence we have $$\mbox{rank}\ \widetilde{A_{n}(Q)} \leq |\Sigma_n^1|.$$

As Tanabe showed in [@Ta], $\Sigma_n^1$ makes a basis of ${\mbox{${\mathbb C}$}}\otimes A_{n}(k) = {\mbox{${\mathbb C}$}}\otimes \psi(\widetilde{A_{n}(k)})$ if $k\geq n$. Hence $\mbox{rank}\ {\mbox{${\mathbb C}$}}\otimes A_{n}(z) = |\Sigma_n^1|$ holds as far as $z$ takes any integer value $k\geq n$. This implies that $\psi$ is an isomorphism and we find that the generators and the relations in Theorem 1.2 characterize the partition algebra $A_{n}(Q)$.

Definition of $A_{n-\frac{1}{2}}(Q)$, a subalgebra of $A_n(Q)$ {#sec:5-1}
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In this section, we consider a subalgebra $A_{n-\frac{1}{2}}(Q)$ of $A_n(Q)$ generated by the special elements $s_1, \ldots, s_{n-2}$, $f_1, \ldots, f_{n-1}$ and $e_1,\ldots, e_{n-1}$. As we have noted in Remark \[rem:gen\], $\{f_i\}$ ($1\leq n-2$) and $\{e_i\}$ ($1\leq n-1$) are written as products of $f=f_1$, $e = e_1$ and $s_1,\ldots, s_{n-2}$. The special element $f_{n-1}$, however, can not be expressed as a product of other special elements in $A_{n-\frac{1}{2}}(Q)$, since we deleted $s_{n-1}$ from the generators of $A_n(Q)$. Hence $A_{n-\frac{1}{2}}(Q)$ can be defined as a subalgebra of $A_n(Q)$ generated by the following elements: $s_1, \ldots, s_{n-2}$, $f = f_1$, $f_* = f_{n-1}$ and $e = e_1$. We can obtain the defining relations among these generators just as in the case of $A_n(Q)$.

\[def:half-int-alg\] Let $\mathbb{Z}$ be the ring of rational integers and $Q$ the indeterminate. We put ${A}_{\frac{1}{2}}(Q) = \mathbb{Z}[Q]\cdot 1$. For an integer $n\geq2$, ${A}_{n-\frac{1}{2}}(Q)$ is characterized by the generators $$e, f, s_1, s_2, \ldots, s_{n-2}, f_{*} \mbox{(if $n>2$)}$$ and the relations ($R0$), ($R1'$)-($R4'$) and ($