Question: A partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_s)$ is a tuple of positive integers listed in non decreasing order such that $\sum_{i=1}\lambda_{i}=N.$ Each $\lambda_{i}$ is called a part of the partition. I write $\mathcal{P}(N)$ for the set of all partitions of $N.$ I write $\leq$ for the dominance order on $\mathcal{P}(n).$ Indeed $\leq$ induces a partial order on $\mathcal{P}(N).$ Brylawski has shown that $(\mathcal{P}(N),\leq)$ is a lattice; which for shorthand is normally denoted by $L(N).$ Brylawski gives the following formulas for $N>1$ of the maximum number of elements covered by any element of $L(N):$ $$ \max_{L(N)}N=\bigg\lfloor-\frac{3}{2}+\frac{1}{2}\sqrt{8N+9}\bigg\rfloor \label{a}\tag{1} $$ Consider the binary string $$B=010010001000010000010000001\ldots$$ I write $B(N)$ for the first $N$ bits of $B.$ I write $\chi_{w}(B(N))$ to count occurrences of the string $w$ up to and including the first $N$ bits of $B.$ Now, suppose $w=1$ then calculating $\chi_{1}(B(N))$ is the same as determining the Hamming weight of $B(N).$ Curiously $$ \chi_{1}(B(N))=\bigg\lfloor-\frac{3}{2}+\frac{1}{2}\sqrt{8N+9}\bigg\rfloor \label{b}\tag{2} $$ What if any are the intuitions that might relate partitions under dominance ordering to hamming weights in our binary string $B?$
By Taking $w=01$ the formula for $\chi_{10}(B(N))$ is the exact formula given by Brylawski for $N\geq1.$ The paper by Brylawski can be found on Science Direct: The lattice of integer partitions Corollary 2.10 -Thomas Brylawski. Also worth mentioning are the partial sums of $1$ and $2$ which are related to the sequence A023535.