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We let $A=R/I$ be a standard graded Artinian algebra quotient of $R={\sfk}[x,y]$, the polynomial ring in two variables over a field ${\sf k}$ by anideal $I$, and let $n$ be its vector space dimension. The Jordan type $P_\ell$of a linear form $\ell\in A_1$ is the partition of $n$ determining the Jordanblock decomposition of the multiplication on $A$ by $\ell$ -- which isnilpotent. The first three authors previously determined which partitions of$n=\dim_{\sf k}A$ may occur as the Jordan type for some linear form $\ell$ on agraded complete intersection Artinian quotient $A=R/(f,g)$ of $R$, and theycounted the number of such partitions for each complete intersection Hilbertfunction $T$ arXiv:1810.00716.\par  We here consider the family $\mathrm{G}_T$ of graded Artinian quotients$A=R/I$ of $R={\sf k}[x,y]$, having arbitrary Hilbert function $H(A)=T$. TheJordan cell $\mathbb V(E_P)$ corresponding to a partition $P$ having diagonallengths $T$ is comprised of all ideals $I$ in $R$ whose initial ideal is themonomial ideal $E_P$ determined by $P$. These cells give a decomposition of thevariety $\mathrm{G}_T$ into affine spaces. We determine the generic number$\kappa(P)$ of generators for the ideals in each cell $\mathbb V(E_P)$,generalizing a result of arXiv:1810.00716. In particular, we determine thosepartitions for which $\kappa(P)=\kappa(T)$, the generic number of generatorsfor an ideal defining an algebra $A$ in $\mathrm{G}_T$. We also count thenumber of partitions $P$ of diagonal lengths $T$ having a given $\kappa(P)$. Amain tool is a combinatorial and geometric result allowing us to split $T$ andany partition $P$ of diagonal lengths $T$ into simpler $T_i$ and partitions$P_i$, such that $\mathbb V(E_P)$ is the product of the cells $\mathbbV(E_{P_i})$, and $T_i$ is single-block: $\mathrm{G}_{T_i}$ is a Grassmannian.

Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two