The fourteen elements below (1 2 3 4), the noncrossing partitions of a square, and their Kreweras complements

(The fourteen interval elements.) By [F3], ${\mathrm{\ell }}_T(w)=4-\mathrm{\#}\{\text{cycles of }w\}$ and $w{\le }_{T}c$ exactly when its cycles are cyclically increasing and its support partition is noncrossing. Length $0$ gives only $1$. Length $1$ gives all six transpositions; each has one pair block and two singleton blocks, so is noncrossing, and its 2-cycle is cyclically increasing. Length $2$ means two cycles, hence either a 3-cycle and a fixed point or two transpositions. For each of the four 3-element supports, exactly one orientation is cyclically increasing, giving $(123),(124),(134),(234)$; each support partition is noncrossing. Of the three double transpositions, $(12)(34)$ and $(14)(23)$ have noncrossing pair blocks, while $(13)(24)$ has crossing pair blocks. Length $3$ means a single 4-cycle; only $(1234)$ is cyclically increasing in the stated order. These cases exhaust the possible cycle counts and give precisely the list in (1).

(Direct complement products.) For an involution $w$, $K(w)=wc$. Applying $c=(1234)$ on the right first gives $K(12)=(234)$, $K(23)=(134)$, $K(34)=(124)$, $K(13)=(12)(34)$, $K(24)=(14)(23)$, and $K(14)=(123)$. The same multiplication gives $K((12)(34))=(24)$ and $K((14)(23))=(13)$. For the 3-cycles, multiplying their inverses by $c$ gives $K(123)=(34)$, $K(124)=(23)$, $K(134)=(12)$, and $K(234)=(14)$; also $K(1)=c$ and $K(c)=1$. This is the full list in Statement (3).

(The fifteen partitions.) By block sizes, the set partitions of four labels consist of one partition with one block, six with three blocks, seven with two blocks, and one with four blocks, for a total of fifteen. A partition with one or four blocks is noncrossing. The six three-block partitions have one pair and two singletons, so are noncrossing. Among the seven two-block partitions, the four triple-plus-singleton partitions are noncrossing; the three pairings are $\{1,2\}\mid \{3,4\}$, $\{1,4\}\mid \{2,3\}$, and $\{1,3\}\mid \{2,4\}$, of which only the last has alternating endpoints. This proves the unique crossing claim. The first-step list has fourteen elements, all with noncrossing cyclically increasing cycles; [F3] says each noncrossing partition has a unique such interval permutation. Thus the supports in (1) give exactly the fourteen noncrossing partitions.

(Order, square, and block counts.) [F4] gives that $K$ is an order-reversing bijection of the interval, $K^2(w)=c^{-1}wc$, and ${\mathrm{\ell }}_T(K(w))=3-{\mathrm{\ell }}_T(w)$. Conjugating a cycle by $c^{-1}$ relabels each entry by $1\mapsto 4\mapsto 3\mapsto 2\mapsto 1$, so the support partition rotates as stated. Since ${\mathrm{\ell }}_T(w)=4-\mathrm{\mid }\pi (w)\mathrm{\mid }$ and ${\mathrm{\ell }}_T(K(w))=4-\mathrm{\mid }\pi (K(w))\mathrm{\mid }$ by [F3], the length complement gives $\mathrm{\mid }\pi (K(w))\mathrm{\mid }=5-\mathrm{\mid }\pi (w)\mathrm{\mid }$.