Harmonic morphisms to P_4 – examples

This post expands on a previous post and gives more examples of harmonic morphisms to the path graph \Gamma_2=P_4.
path4-0123

We only consider the cyclic graph on k vertices, C_k as the domain in this post. There are no non-trivial harmonic morphisms C_5 \to P_4, so we start with C_6. We indicate a harmonic morphism by a vertex coloring. An example of a harmonic morphism can be described in the plot below as follows: \phi:\Gamma_1\to \Gamma_2=P_4 sends the red vertices in \Gamma_1 to the red vertex of \Gamma_2=P_4 (we let 3 be the numerical notation for the color red), the blue vertices in \Gamma_1 to the blue vertex of \Gamma_2=P_4 (we let 2 be the numerical notation for the color blue), the green vertices in \Gamma_1 to the green vertex of \Gamma_2=P_4 (we let 1 be the numerical notation for the color green), and the white vertices in \Gamma_1 to the white vertex of \Gamma_2=P_4 (we let 0 be the numerical notation for the color white).

To get the following data, I wrote programs in Python using SageMath.

Example 1: There are only the 4 trivial harmonic morphisms C_6 \to P_4, plus that induced by f = (1, 2, 3, 2, 1, 0) and all of its cyclic permutations (4+6=10). This set of 6 permutations is closed under the automorphism of P_4 induced by the transposition (0,3)(1,2) (so total = 10).cyclic6-123210

Example 2: There are only the 4 trivial harmonic morphisms, plus f = (1, 2, 3, 2, 1, 0, 0) and all of its cyclic permutations (4+7=11). This set of 7 permutations is not closed under the automorphism of P_4 induced by the transposition (0,3)(1,2), so one also has f = (2, 1, 0, 1, 2, 3, 3) and all 7 of its cyclic permutations (total = 7+11 = 18).
cyclic7-1232100
cyclic7-1233210

Example 3: There are only the 4 trivial harmonic morphisms, plus f = (1, 2, 3, 2, 1, 0, 0, 0) and all of its cyclic permutations (4+8=12). This set of 8 permutations is not closed under the automorphism of P_4 induced by the transposition (0,3)(1,2), so one also has f = (1, 2, 3, 3, 3, 2, 1, 0) and all of its cyclic permutations (12+8=20). In addition, there is f = (1, 2, 3, 3, 2, 1, 0, 0) and all of its cyclic permutations (20+8 = 28). The latter set of 8 cyclic permutations of (1, 2, 3, 3, 2, 1, 0, 0) is closed under the transposition (0,3)(1,2) (total = 28).
cyclic8-12321000
cyclic8-12333210
cyclic8-12332100

Example 4: There are only the 4 trivial harmonic morphisms, plus f = (1, 2, 3, 2, 1, 0, 0, 0, 0) and all of its cyclic permutations (4+9=13). This set of 9 permutations is not closed under the automorphism of P_4 induced by the transposition (0,3)(1,2), so one also has f = (1, 2, 3, 3, 2, 1, 0, 0, 0) and all 9 of its cyclic permutations (9+13 = 22). This set of 9 permutations is not closed under the automorphism of P_4 induced by the transposition (0,3)(1,2), so one also has f = (1, 2, 3, 3, 3, 2, 1, 0, 0) and all 9 of its cyclic permutations (9+22 = 31). This set of 9 permutations is not closed under the automorphism of P_4 induced by the transposition (0,3)(1,2), so one also has f = (1, 2, 3, 3, 3, 3, 2, 1, 0) and all 9 of its cyclic permutations (total = 9+31 = 40). cyclic9-123210000cyclic9-123321000cyclic9-123332100cyclic9-123333210

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