Appendix

In the following C^{7} denotes a chain of rank 7

A.1 P (6, {[1, 2], [2, 6]}) + P (6, {[2, 5], [5, 6]})

Take D^{↔} _{[1,2]}^{N}D^{↔} _{[2,6]}^{N} (C^{7}) and D^{↔} _{[1,5]}^{N}D^{↔} _{[5,6]}^{N} (C^{7}). Identify the elements of both posets at rank 1 and at rank 6. Figure 1 represents the resulting poset for N = 2.

[figure: .] Figure 1: P (6, {[1, 2], [2, 6]}) + P (6, {[2, 5], [5, 6]})

A.2 P (6, {[1, 3], [3, 4], [4, 6]}) + P (6, {[1, 2], [2, 3]}) + P (6, {[4, 5], [5, 6]})

Take

P^{I}(N) = D^{↔} _{[1,3]}^{N}D^{↔} _{[3,4]}^{N}D^{↔} _{[4,6]}^{N}D^{↔} _{[4,5]}^{N+1} (C^{7})
P^{II}(N) = D^{↔} _{[1,2]}^{N+1}D^{↔} _{[1,6]}^{N}D^{↔} _{[2,4]}^{N} (C^{7}) , and
P^{III}(N) = D^{↔} _{[1,5]}^{N}D^{↔} _{[3,5]}^{N}D^{↔} _{[5,6]}^{N} (C^{7}) .

Identify the elements of P^{I}(N) with the elements of P^{II}(N) at ranks 1, 4, 5, and 6. Identify the elements of P^{I}(N) with the elements of P^{III}(N) at ranks 1, 2, 3, and 6. Figure 2 represents the resulting poset for N = 2.

[figure] Figure 2: P (6, {[1, 3], [3, 4], [4, 6]}) + P (6, {[1, 2], [2, 3]}) + P (6, {[4, 5], [5, 6]})

A.3 P (6, {[1, 2], [3, 4], [4, 5]}) + P (6, {[3, 5], [5, 6]}) + P (6, {[1, 2], [2, 5]})

Take

P^{I}(N) = D^{↔} _{[1,2]}^{N+1}D^{↔} _{[3,4]}^{N+1}D^{↔} _{[3,6]}^{N}D^{↔} _{[4,5]}^{N+1} (C^{7}) (Figure 3)
P^{II}(N) = D^{↔} _{[1,5]}^{N+1}D^{↔} _{[3,5]}^{N^{2}}D^{↔} _{[5,6]}^{N} (C^{7})  (Figure 4), and
P^{III}(N) = D^{↔} _{[1,2]}^{N+2}D^{↔} _{[2,5]}^{N^{2}−N+2}D^{↔} _{[1,6]}^{N} (C^{7}) (Figure 5).

Identify the elements of P^{I}(N) with the elements of P^{II}(N) at ranks 1, 2, and 6. Identify the elements of P^{I}(N) with the elements of P^{III}(N) at rank 6. Figure 6 represents the resulting poset for N = 2.

[figure] Figure 3: P^{I}(N)

[figure: Figure 4: P^{II}(N)]

4 [figure: Figure 5: P^{III}(N)]

5 [figure: Figure 6: P (6, {[1, 2], [3, 4], [4, 5]}) + P (6, {[3, 5], [5, 6]}) + P (6, {[1, 2], [2, 5]})]

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