  We investigate partitioning of integer sequences into heapable subsequences
(previously defined and established by Mitzenmacher et al). We show that an
extension of patience sorting computes the decomposition into a minimal number
of heapable subsequences (MHS). We connect this parameter to an interactive
particle system, a multiset extension of Hammersley's process, and investigate
its expected value on a random permutation. In contrast with the (well studied)
case of the longest increasing subsequence, we bring experimental evidence that
the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2}\cdot \ln(n)$. Finally
we give a heap-based extension of Young tableaux, prove a hook inequality and
an extension of the Robinson-Schensted correspondence.
