A note on Diagonal sequences of integer partitions

Let $\mathcal{P}(n)$ be the set of partitions of the positive integer $n$. For $\alpha =({\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_t)\in \mathcal{P}(n)$ define the diagonal sequence $\delta (\alpha )=(d_k(\alpha ){)}_{k\ge 1}$ via $d_k(\alpha )=\mid \{i\mid 1\le i\le kand{\alpha }_i+i-1\ge k\}\mid \mathrm{.}$ We show that the set of all partitions in $\mathcal{P}(n)$ with the same diagonal sequence is a partially ordered set under majorization with unique maximal and minimal elements and we give an explicit formula for the number of partitions with the same diagonal sequence.