minicomplexity
 every string of numbers from $[h]$ which contains all numbers of $[h]$
 every pair of $h$-tall $2$-column graphs of rightward arrows which contain a path from their left to their right column
 every pair of $h$-tall $2$-column graphs which contain a path from their left to their right column
 every pair of disjoint subsets of $[h]$
 every binary string with identical $h$-long prefix and suffix
 every pair of equal subsets of $[h]$
 every string of $h$-tall $2$-column graphs of non-merging rightward arrows which contain a path from their leftmost to their righmost column
 every pair of partial functions on $[h]$ which return $0$ on $0$ when composed
 every pair of functions on $[h]$ which contain a cycle
 every pair of functions on $[h]$ and pair of points in $[h]$ such that the functions connect the points
 every pair of functions on $[h]$ which contain a cycle through $0$
 every binary string which can be split into blocks of the form $\mathtt{{1}}(\mathtt{{0}}^*\mathtt{{1}})^{{{\lt}}h-1}\mathtt{{0}}^+$ or $\mathtt{{1}}(\mathtt{{0}}^*\mathtt{{1}})^{h-1}$
 every pair of subsets of $[h]$ the first of which is included in the second one
 every unary string of length a nonzero multiple of $h$
 every unary string of length a nonzero multiple of $2^h$
 every binary string whose $i.h$-th rightmost bit is $\mathtt{{1}}$, for some $i$
 every unary string whose length is a multiple of every summand in a partition of $h$ with maximum least common multiple
 the unique short unary string of length $h$
 every number and list of numbers from $[h]$ such that the number appears in the list
 the unique long unary string of length $2^h$
 every number and set of numbers from $[h]$ such that the number is in the set
 every monotone program of operations on a subset of $[h]$ in which $0$ at start implies $h{-}1$ at finish
 every binary string which can be split into blocks of the form $\mathtt{{0}}^*\mathtt{{1}}((\mathtt{{0}}{{+}}\mathtt{{1}})^{h-2}\mathtt{{1}})^+$
 every string of $h$-tall $2$-column graphs of rightward arrows which contain a path from their leftmost to their righmost column
 every number and ordered list of numbers from $[h]$ such that the number appears in the list
 every binary string which can be split into blocks of the form $\mathtt{{1}}(\mathtt{{0}}{{+}}\mathtt{{1}})^*\mathtt{{0}}$ or $\mathtt{{1}}((\mathtt{{0}}^*\mathtt{{1}})^{h-1})^+$
 every number, index, and tuple from $[h]$ such that the number is at the indexed entry of the tuple
 every pair of binary relations on $[h]$ which contain a cycle
 every pair of binary relations on $[h]$ and pair of points in $[h]$ such that the relations connect the points
 every pair of binary relations on $[h]$ which contain a cycle through $0$
 every binary string whose $h$-th rightmost bit is $\mathtt{{1}}$
 every string of subsets of $[h]$ which can be split into blocks so that the first set in each block contains the number of sets after it
 every $0$-preserving program of operations on a subset of $[h]$
 every short string of numbers from $[h]$ which contains all numbers of $[h]$
 every short binary string with identical $h$-long prefix and suffix
 every short binary string whose $h$-th rightmost bit is $\mathtt{{1}}$
 every string of $h$-tall $2$-column graphs which contain a path from their leftmost to their rightmost column
 every unary string whose length is a multiple of some nontrivial summand in a partition of $h$ with maximum least common multiple