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Figure 1.
Let $ x\in X_n $ be a combinatorial structure on $ n $ vertices. $ A_x $ choose a permutation $ \sigma $ depending on $ x $ and permutes the vertices of $ x $ according to it. We represent this symmetric-group action as a diagrammatic action from below
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Figure 2.
$ S' $ is a boosted sequence of $ S $ for $ (x, c) $, because $ \text{red} \circ S' $ gives the same result as $ S \circ \text{red} $. Note that $ S' $ may be not unique: in the diagram we show a second boosted sequence of $ S $ for $ (x, c) $, namely $ S" $. It is not necessarily the case that $ (x', c') : = S'(x, c) $ coincides with $ (x", c") : = S"(x, c) $, but merely that their reductions coincide (they must be both $ y' $)
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Figure 3.
A representation of a combinatorial structure $ x\in X_n $ with a labelling $ \Pi_b. $ The intervals are represented by the bottoms arcs
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Figure 4.
Outline of the proof of connectivity between $ x_1 $ and $ x_2 $, using the labelling method. The sequence $ S $ sends $ y_1 $ to $ y_2 $, however the intervals containing the gray vertices of $ x'_1 $ may not be at their correct place, in order to match with those of $ x'_2 $. The sequence $ S_* $ corrects for this. Thus the boosted sequence $ S'_* S' $ sends $ x'_1 $ to $ x'_2 $, and $ x_1 $ and $ x_2 $ are connected
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Figure 5.
Diagram representations of matchings and permutations, and matrix representation of permutations
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Figure 6.
Our two main examples of dynamics, in diagram representation. Top: the $ \mathcal{M}_n $ case. top: the $ \mathcal{S}_n $ case
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Figure 7.
The Rauzy dynamics concerning permutations, in matrix representation
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Figure 8.
Left: A reducible permutation in matrix representation; the $ \mathcal{S} $ dynamic acts on the first block with $ L $ and $ R $. Right: A reducible permutation in diagram representation; the $ \mathcal{S} $ dynamics acts on the gray part with $ L $ and $ R $, while leaving the blue part unchanged
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Figure 9.
Left: an irreducible permutation, $ \sigma = [\, 451263\, ] $. Right: the construction of the cycle structure. Different cycles are in different colours. The length of a cycle or path, defined as the number of top (or bottom) arcs, is thus 2 for red and violet, and 1 for blue. The green arrows are the endpoints of the rank path, which is in blue. As a result, in this example $ \lambda (\sigma ) = (2, 2) $ (for the cycles of color red and violet), $ r(\sigma ) = 1 $ (corresponding to the rank path of color blue), and $ \ell(\sigma ) = 2 $
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Figure 10.
Left: an example of permutation, $ \sigma = [\, 251478396\, ] $. Right: an example of subset $ I = \{1, 2, 6, 8, 9\} $ (labels are for the bottom endpoints, edges in $ I $ are in blue). There are two crossings, out of the maximal number $ \binom{|I|}{2} = 10 $, thus $ \chi_I = 8 $ in this case, and this set contributes $ (-1)^{|I|+\chi_I} = (-1)^{5+8} = -1 $ to $ A(\sigma ) $
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Figure 11.
Two consistent labellings $ (\Pi_b, \Pi_t) $ and $ (\Pi'_b, \Pi'_t) $ of a permutation $ \sigma $ with cycle invariant $ (\{2, 2, 2\}, 2) $. Following definition 26 we have $ (\Pi'_b, \Pi'_t) = \mathrm{Sh}^1_{2, 1}((\Pi_b, \Pi_t)) $
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Figure 12.
The action on the dynamics (with an $ L $ operator) on a consistent labelling
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Figure 13.
The proof that $ \Pi'_b $ verifies property 2 and $ (\Pi'_b, \Pi'_t) $ verifies property 3 for the case $ \alpha\leq \sigma (1) $
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Figure 14.
The proof that $ \Pi'_b $ verifies property 2 and $ (\Pi'_b, \Pi'_t) $ verifies property 3 for the case $ \alpha = \sigma (1)+1 $
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Figure 15.
The gray edges of $ \sigma $ are transported along the labels of the arcs when applying the boosted dynamics. For instance, $ e\in(b, t) $ in $ \tau, \Pi $
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Figure 16.
Illustration that the edges, inserted within the labels of a reduced permutation, follow those labels when applying the boosted dynamic
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Figure 17.
Left: a schematic representation of a permutation of type $ H(r_1, r_2) $. Right: a representation of a permutation of type $ X(r, i) $. These configurations have rank $ r_1+r_2-1 $ and $ r $, respectively
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Figure 18.
The insertion of one edge within the arcs $ \alpha $ and $ \beta $
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Figure 19.
The first line represents the case: Top arc : any cycle. Bottom arc: any cycle. The second line represents the case: Top arc : rank path. Bottom arc: principal cycle
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Figure 20.
The first column represents the permutation $ \sigma $ and its the cycle invariant. The second column represents the permutation $ \sigma ' $ obtained from the insertion of a double-edge within the arcs $ \alpha $ and $ \beta $ of $ \sigma $ and the third column displays and demonstrates the new invariant of $ \sigma ' $
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Figure 21.
An example of a shift-irreducible family $ S = (\sigma ^i)_{0\leq i \leq n-2} $ with its two unavoidably reducible permutations $ d(\sigma ^{n-\sigma (2)}) $ and $ d(\sigma ^{n-\sigma (n)+1}). $ For a proof of the shift-irreducibility, the characterisation proposition 44 is helpful
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Figure 22.
The two permutations $ \sigma ^{n-\sigma (2)} $ and $ \sigma ^{n-\sigma (n)+1} $ are of type $ H(1, r) $ and $ H(r, 1) $ respectively
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Figure 23.
The permutation $ \sigma _{1, 4, \{(0.1, 0.1), (0.2, 3.1), (0.3, 1.1), (1.1, 4.1), (1.2, 2.1)\}} $. We cannot show the full matrix $ Q $ for such a big example, but we can give one row, for the edge which has the label $ e $ in the drawing. The row $ Q_e $ reads $ (Q_e)_{11, 12, \ldots, 15, 21, \ldots, 25} = (1, 1, 0, 0, 0, \, 0, 0, 1, 1, 1) $
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Figure 24.
From left to right: the permutations $ \sigma $, $ \sigma |_{1, \alpha, \beta} $ and $ \sigma |_{1, \alpha, \beta'} $. The proposition 53 says that $ \sigma |_{1, \alpha, \beta} $ and $ \sigma |_{1, \alpha, \beta'} $ have same cycle invariant and opposite sign invariant
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Figure 25.
Left: a permutation $ \sigma $. Right : the permutation $ \sigma _i(C_p) $
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Figure 26.
The base cases of the induction of the proof of proposition 58
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Figure 27.
Left: a permutation with a $ C_p $ structure containing two even cycles of even length. Middle and Right: By proposition 53 the two permutations have same cycle invariant and opposite sign
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Figure 28.
Two families of base permutations with their respective cycle invariant
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Figure 29.
The permutation resulting from adding a $ C_p $ on the last edge of $ \sigma ^0 $ or one of its successors. It is clearly also $ I_2X $
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Figure 30.
The two constructed $ I_2X $-permutations $ \sigma _1 = \sigma _{|\sigma |}(C_p) $ and $ \sigma '_1 = \sigma _{|\sigma |}(C_{p-(2), 2}) $ with invariant $ (\lambda , r, \pm s) $. In this case $ p = 4k+1 $
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Figure 31.
The scheme to add two different even cycles to a permutation
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Figure 32.
Two families of $ I_2X $-permutations with their respective cycle invariant
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Figure 33.
The case $ \sigma $ has type $ X(r, i) $ of lemma 68. We have $ \sigma _1 = \tau_{1, \alpha_1 = 1, \beta} = \tau_{1, t^{rk}_0, b^{rk}_{i\!-\!1}} $
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Figure 34.
The case $ \sigma $ has type $ H(i, r_2) $ of lemma 69. We have $ \sigma _1 = \tau_{1, \alpha_1 = 1, \beta} = \tau_{1, t^{rk}_0, b_{i-1, i\!-\!1, 1}} $
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Figure 35.
The construction of the proof of the lemma. We have $ \sigma _1 = \tau_{1, 1, \beta'} $, $ \sigma _1' = \tau_{1, 1, \beta} $ and $ \sigma _1 = B(S)(\sigma _1) $ moreover $ {\bar A}(\tau) = \pm 2^{\frac{n}{2}} $
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Figure 36.
The construction of the proof of the lemma 73. Both $ \sigma $ and $ \sigma _{next} $ have type $ X(r, i) $ and $ {\bar A}(d(\sigma )) = -{\bar A}(d(\sigma _{next})) $
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Figure 37.
$ (\sigma '_4, \sigma '_2) $ are both permutations with invariant $ (\lambda , r, s\neq 0) $ and $ \tau_2 $ has sign invariant 0. However proposition 19 applied to $ \sigma '_4, \sigma '_2 $ and $ \tau_2 $ implies that $ \sigma '_4 $ should have invariant $ -s $, This is contradictory and thus the case $ \Pi'''_b(\beta) = b_{0, i-1, 1} $ cannot happen
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Figure 38. $ \sigma $ and $ \sigma ' $ are standard of type $ X(r, i) $. The labels of the principal cycle of $ \sigma $ are send to the arcs of the principal cycle of $ \sigma ' $ by the sequence $ R^{-1}B(S)R $. Indeed they are attached to the $ i $th first labels of the rank of $ \tau $ and the labels of the rank are fixed. Thus they are also attached to $ i $th first the labels of the rank of $ \tau' $ which are the arcs of the principal cycle of $ \sigma ' $.
In the figure, we choose $ \Pi_t(\alpha) = t_{0, i, j} $ instead of $ t_{c, i, j} $ for some $ c $ for space-saving purpose
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Figure 39.
The case of the cycle 1-shift. We apply proposition 28 on $ \sigma ' $ and $ \sigma $
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Figure 40.
The case of the cycle jump. We apply proposition 76 on $ \sigma ' $ and $ \sigma $
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Figure 41.
The case of the cycle 2-shift. We apply proposition 28 on $ \sigma ' $ and $ \sigma $
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Figure 42.
Description of the structure of configurations in the identity class $ {\rm{Id}}_n $, besides the configuration $ {\rm{id}}_n $
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