  
  [1X9 [33X[0;0YRegular Languages of Sets of Permutations[133X[101X
  
  [33X[0;0YThis  chapter  is  dedicated  to the different sets of permutations with the
  same properties.[133X
  
  
  [1X9.1 [33X[0;0YInversions in Permutations[133X[101X
  
  [33X[0;0YAn  inversion in a permutation [22Xτ=τ_1...τ_n[122X is a pair [22X(i,j)[122X such that [22X1≤ i<j≤
  n[122X and [22Xτ_i>τ_j[122X [5].[133X
  
  [1X9.1-1 InversionAut[101X
  
  [33X[1;0Y[29X[2XInversionAut[102X( [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn   automaton  that  accepts  all  permutations  with  exactly  [10Xk[110X
            inversions.[133X
  
  [33X[0;0YThe  rational  language  of  all  permutations  with  a given number , [10Xk[110X, of
  inversions is computed by [10XInversionAut[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=InversionAut(1);[127X[104X
    [4X[28X< deterministic automaton on 2 letters with 4 states >[128X[104X
    [4X[25Xgap>[125X [27XAutToRatExp(a);[127X[104X
    [4X[28Xa*baa*[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);     [127X[104X
    [4X[28X[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ][128X[104X
    [4X[25Xgap>[125X [27Xb:=InversionAut(5);[127X[104X
    [4X[28X< deterministic automaton on 6 letters with 14 states >[128X[104X
    [4X[25Xgap>[125X [27XAutToRatExp(b);[127X[104X
    [4X[28X((a*ba*bUa*c)a*bUa*ba*cUa*d)a*(ba*baa*Ucaaa*)U(a*ba*bUa*c)a*(ca*baa*Udaaaa*)U(\[128X[104X
    [4X[28Xa*ba*daUa*eaa)a*baa*Ua*ba*(dbUeaa)aaa*U(a*eabUa*(ebUfaa)a)aaa*[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(b);   [127X[104X
    [4X[28X[ 0, 0, 0, 3, 22, 71, 169, 343, 628, 1068, 1717, 2640, 3914, 5629, 7889 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.1-2 InversionAutOfClass[101X
  
  [33X[1;0Y[29X[2XInversionAutOfClass[102X( [3Xaut[103X, [3Xinv[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn  automaton  accepting  all  permutations  of  a  class with [10Xinv[110X
            inversions.[133X
  
  [33X[0;0Y[10XInversionAutOfClass[110X  intersects the rational pattern class with the rational
  language  containing  all  permutations  under  the  rank encoding that have
  exactly [10Xinv[110X inversions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=MinimalAutomaton(GraphToAut(Seqstacks(2,2),1,6));[127X[104X
    [4X[28X< deterministic automaton on 3 letters with 3 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);                                        [127X[104X
    [4X[28X[ 1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, [128X[104X
    [4X[28X  1062882, 3188646 ][128X[104X
    [4X[25Xgap>[125X [27Xb:=InversionAutOfClass(a,4);                        [127X[104X
    [4X[28X< deterministic automaton on 5 letters with 23 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(b);                                        [127X[104X
    [4X[28X[ 0, 0, 0, 3, 13, 35, 75, 140, 238, 378, 570, 825, 1155, 1573, 2093 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  
  [1X9.2 [33X[0;0YPlus- and Minus-(In)Decomposablilty[133X[101X
  
  [1X9.2-1 PlusDecomposableAut[101X
  
  [33X[1;0Y[29X[2XPlusDecomposableAut[102X( [3Xaut[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn  automaton  that accepts the subset of the class [10Xaut[110X containing
            the plus-decomposable permutations of [10Xaut[110X.[133X
  
  [33X[0;0YThe    [10XPlusDecomposableAut[110X   automaton   accepts   the   language   of   all
  plus-decomposable permutations of the encoded class accepted by [10Xaut[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xxa:=MinimalAutomaton(GraphToAut(Parstacks(2,2),1,6));[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 9 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(xa);[127X[104X
    [4X[28X[ 1, 2, 6, 23, 89, 345, 1338, 5189, 20122, 78024, 302529, 1172993, 4547973, [128X[104X
    [4X[28X  17633432, 68368135 ][128X[104X
    [4X[25Xgap>[125X [27Xa:=PlusDecomposableAut(xa);[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 16 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);[127X[104X
    [4X[28X[ 0, 1, 3, 11, 47, 196, 808, 3306, 13433, 54265, 218145, 873303, 3483654, [128X[104X
    [4X[28X  13853682, 54945158 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.2-2 PlusIndecomposableAut[101X
  
  [33X[1;0Y[29X[2XPlusIndecomposableAut[102X( [3Xaut[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn  automaton  that  accepts  all permutations of [10Xaut[110X that are not
            plus-decomposable.[133X
  
  [33X[0;0YThe  [10XPlusIndecomposableAutomaton[110X  automaton  accepts  the  language  of  all
  plus-indecomposable  permutations  of  the encoded class accepted by aut, by
  rejecting  every  rank  encoding  that  in the original automaton would have
  entered  and  left  the  accept  state  before  the  last letter in the rank
  encodedpermutation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xxa:=MinimalAutomaton(GraphToAut(Parstacks(2,2),1,6));[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 9 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(xa);[127X[104X
    [4X[28X[ 1, 2, 6, 23, 89, 345, 1338, 5189, 20122, 78024, 302529, 1172993, 4547973, [128X[104X
    [4X[28X  17633432, 68368135 ][128X[104X
    [4X[25Xgap>[125X [27Xa:=PlusIndecomposableAut(xa);[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 11 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);[127X[104X
    [4X[28X[ 1, 1, 3, 12, 42, 149, 530, 1883, 6689, 23759, 84384, 299690, 1064319, [128X[104X
    [4X[28X  3779750, 13422977 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.2-3 MinusDecomposableAut[101X
  
  [33X[1;0Y[29X[2XMinusDecomposableAut[102X( [3Xaut[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn  automaton  that accepts the subset of the class [10Xaut[110X containing
            the minus-decomposable permutations of [10Xaut[110X.[133X
  
  [33X[0;0YThe   [10XMinusDecomposableAut[110X   automaton   accepts   the   language   of   all
  minus-decomposable permutations of the rank encoded class accepted by [10Xaut[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xxa:=MinimalAutomaton(GraphToAut(Parstacks(2,2),1,6));[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 9 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(xa);[127X[104X
    [4X[28X[ 1, 2, 6, 23, 89, 345, 1338, 5189, 20122, 78024, 302529, 1172993, 4547973, [128X[104X
    [4X[28X  17633432, 68368135 ][128X[104X
    [4X[25Xgap>[125X [27Xa:=MinusDecomposableAut(xa);                         [127X[104X
    [4X[28X< deterministic automaton on 4 letters with 12 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);                                         [127X[104X
    [4X[28X[ 0, 1, 3, 10, 24, 64, 180, 520, 1524, 4504, 13380, 39880, 119124, 356344, [128X[104X
    [4X[28X  1066980 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.2-4 MinusIndecomposableAut[101X
  
  [33X[1;0Y[29X[2XMinusIndecomposableAut[102X( [3Xaut[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn  automaton  that  accepts  all permutations of [10Xaut[110X that are not
            minus-decomposable.[133X
  
  [33X[0;0YThe   [10XMinusIndecomposableAut[110X   automaton   accepts   the   language  of  all
  minus-indecomposable  permutations  of  the  encoded  class accepted by aut,
  which  is  the  complement set of the set of minus-decomposable permutations
  within the class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xxa:=MinimalAutomaton(GraphToAut(Parstacks(2,2),1,6));[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 9 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(xa);[127X[104X
    [4X[28X[ 1, 2, 6, 23, 89, 345, 1338, 5189, 20122, 78024, 302529, 1172993, 4547973, [128X[104X
    [4X[28X  17633432, 68368135 ][128X[104X
    [4X[25Xgap>[125X [27Xa:=MinusIndecomposableAut(xa);[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 17 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);[127X[104X
    [4X[28X[ 1, 1, 3, 13, 65, 281, 1158, 4669, 18598, 73520, 289149, 1133113, 4428849, [128X[104X
    [4X[28X  17277088, 67301155 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  
  [1X9.3 [33X[0;0YLanguage of all non-simple permutations[133X[101X
  
  [33X[0;0YThe  regular  language  of  all  non-simple  rank  encoded permutations with
  highest rank [22Xk[122X is described by the following equation,[133X
  
  
  [24X[33X[0;6YE(NS_k) = E(Ω_k) ∩ ( ⋃_l=1^k-1 P_l ⋃_m=l^k-1 E(hatΩ_k-m)^+m ∪ ⋃_j=1^k-1 E(hatΩ_k-j)^+j ∪[133X[124X
  
  
  [24X[33X[0;6Y∪ ⋃_a=2^k-1 ⋃_b=0^k-1-a Q_a,b ⋃_i=0^a-2 (E(hatΩ_k-(b+i))^+b+i)^(a-i) ) Σ^* ∪ E(mathcalD_P(Ω_k))[133X[124X
  
  [33X[0;0Ywhere [22XΣ[122X is the alphabet [22X{1,...,k}[122X, [22Xk∈N[122X, [22Xk ≥ 3[122X.[133X
  
  [33X[0;0Y[22XP_l[122X  is  the  language  of  prefixes of rank encoded permutations, where the
  prefix ends with the total sum of gap sizes to be equal to [22Xl[122X.[133X
  
  [33X[0;0Y[22XQ_i,j[122X  is  the  language of prefixes of rank encoded permutations, where the
  prefix  ends  with  a  gap  of size [22Xi[122X and the sum of the sizes of gaps below
  equals to [22Xj[122X.[133X
  
  [33X[0;0Y[22XE(Ω_k-i)^+i[122X  is  the  language  of [22XE(Ω_k-i)[122X [22Xi ∈ N[122X, with the alphabet shifted
  upwards by [22Xi[122X.[133X
  
  [33X[0;0Y[22XE(Ω_k)^(i)[122X  is the sublanguage of [22XE(Ω_k)[122X containing the words of length [22X≤ i[122X,
  [22Xi ∈ N[122X.[133X
  
  [33X[0;0Y[22XE(hatΩ_k)[122X is the sublanguage of [22XE(Ω_k)[122X excluding the words of length [22X≤ 1[122X.[133X
  
  [33X[0;0Y[22XE(mathcalD_P(Ω_k))[122X  is the language of all plus-decomposable permutations as
  described in [6].[133X
  
  [1X9.3-1 LengthBoundAut[101X
  
  [33X[1;0Y[29X[2XLengthBoundAut[102X( [3Xaut[103X, [3Xmin[103X, [3Xi[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe subautomaton of [10Xaut[110X that accepts words between (and including)
            the lengths [10Xmin[110X and [10Xi[110X.[133X
  
  [33X[0;0YWe  are taking the automaton [10Xaut[110X and it's alphabet [10Xk[110X, and find the automaton
  that accepts all words of [10Xaut[110X of length between (and including) [10Xmin[110X and [10Xi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=BoundedClassAutomaton(4); [127X[104X
    [4X[28X< deterministic automaton on 4 letters with 4 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(a);[127X[104X
    [4X[28X[ 1, 2, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, [128X[104X
    [4X[28X  25165824, 100663296 ][128X[104X
    [4X[25Xgap>[125X [27XLengthBoundAut(a,4,8,4);[127X[104X
    [4X[28X< deterministic automaton on 4 letters with 22 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(last);[127X[104X
    [4X[28X[ 0, 0, 0, 24, 96, 384, 1536, 6144, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.3-2 ShiftAut[101X
  
  [33X[1;0Y[29X[2XShiftAut[102X( [3Xi[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe automaton [22XΩ_k-i^+i[122X.[133X
  
  [33X[0;0YWe  are shifting the alphabet of [22XΩ_k-i[122X in their values by [22Xi[122X to expand to the
  alphabet [22X{1,...,k}[122X, but keeping the automaton structure of [22XΩ_k-i[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XShiftAut(2,4);[127X[104X
    [4X[28X< non deterministic automaton on 4 letters with 4 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1       2       3       4[128X[104X
    [4X[28X-----------------------------------[128X[104X
    [4X[28X a |                                 [128X[104X
    [4X[28X b |                                 [128X[104X
    [4X[28X c | [ 2 ]   [ 4 ]   [ 4 ]   [ 4 ]   [128X[104X
    [4X[28X d | [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28XInitial state:   [ 1 ][128X[104X
    [4X[28XAccepting state: [ 4 ][128X[104X
    [4X[25Xgap>[125X [27XShiftAut(1,4);[127X[104X
    [4X[28X< non deterministic automaton on 4 letters with 5 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1       2       3       4       5[128X[104X
    [4X[28X-------------------------------------------[128X[104X
    [4X[28X a |                                         [128X[104X
    [4X[28X b | [ 2 ]   [ 5 ]   [ 5 ]   [ 3 ]   [ 5 ]   [128X[104X
    [4X[28X c | [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X d | [ 4 ]   [ 4 ]   [ 4 ]   [ 4 ]   [ 4 ]   [128X[104X
    [4X[28XInitial state:   [ 1 ][128X[104X
    [4X[28XAccepting state: [ 5 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.3-3 NextGap[101X
  
  [33X[1;0Y[29X[2XNextGap[102X( [3Xgap[103X, [3Xrank[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list of gap sizes.[133X
  
  [33X[0;0YKnowing  the  current  available  gap  sizes  [10Xgap[110X,  which  are the number of
  available  spaces  in a permutation plot. These gaps are separated by blocks
  where  there  are already points inserted. We determine where the next point
  (known to us as its [10Xrank[110X) is being placed and what the next gap sizes are.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNextGap([1,1],2);[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27XNextGap([1],3);[127X[104X
    [4X[28X[ 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XNextGap([2,1],4);[127X[104X
    [4X[28X[ 2, 1 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.3-4 GapAut[101X
  
  [33X[1;0Y[29X[2XGapAut[102X( [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe   non-deterministic   automaton  accepting  the  rank  encoded
            language of [22XΩ_k[122X and the list of all possible gap sizes.[133X
  
  [33X[0;0YThe  automaton  accepts  the  rank  encoded  permutations  of  [22XΩ_k[122X,  but the
  automaton  is slightly extended through having each state corresponding to a
  gap  size  and  the  start  state  being  the  emptyset  of  gap  sizes. The
  transitions  of  the  automaton  are determined through the knowledge of the
  available  spaces  and  the rank. This is calculated in [10XNextGap[110X. Please note
  that  the index of the gap sizes in the list corresponds to the state of the
  automaton.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGapAut(3);[127X[104X
    [4X[28X[ < non deterministic automaton on 3 letters with 5 states >, [128X[104X
    [4X[28X  [ [  ], [ 0 ], [ 1 ], [ 2 ], [ 1, 1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XDisplay(last[1]);[127X[104X
    [4X[28X   |  1       2       3       4       5[128X[104X
    [4X[28X-------------------------------------------[128X[104X
    [4X[28X a | [ 2 ]   [ 2 ]   [ 2 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X b | [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X c | [ 4 ]   [ 4 ]   [ 5 ]   [ 4 ]   [ 5 ]   [128X[104X
    [4X[28XInitial state:    [ 1 ][128X[104X
    [4X[28XAccepting states: [ 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27X [127X[104X
  [4X[32X[104X
  
  [1X9.3-5 SumAut[101X
  
  [33X[1;0Y[29X[2XSumAut[102X( [3Xsum[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe automaton accepting the language [22XP_sum[122X.[133X
  
  [33X[0;0YThis  automaton is based on the [10XGapAut[110X where the accept states are chosen by
  their gap sizes, namely if the total sum of gap sizes equal to [10Xsum[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSumAut(2,3);[127X[104X
    [4X[28X< non deterministic automaton on 3 letters with 5 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1       2       3       4       5[128X[104X
    [4X[28X-------------------------------------------[128X[104X
    [4X[28X a | [ 2 ]   [ 2 ]   [ 2 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X b | [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X c | [ 4 ]   [ 4 ]   [ 5 ]   [ 4 ]   [ 5 ]   [128X[104X
    [4X[28XInitial state:    [ 1 ][128X[104X
    [4X[28XAccepting states: [ 4, 5 ][128X[104X
    [4X[25Xgap>[125X [27X [127X[104X
  [4X[32X[104X
  
  [1X9.3-6 GapSumAut[101X
  
  [33X[1;0Y[29X[2XGapSumAut[102X( [3Xgap[103X, [3Xsum[103X, [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe automaton accepting the language [22XQ_gap,sum[122X.[133X
  
  [33X[0;0YThis  automaton is based on the [10XGapAut[110X where the accept states are chosen by
  their  gap sizes, namely if there is a gap size [10Xgap[110X and the gap sizes before
  have a total sum of [10Xsum[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGapSumAut(1,0,3);[127X[104X
    [4X[28X< non deterministic automaton on 3 letters with 5 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);   [127X[104X
    [4X[28X   |  1       2       3       4       5[128X[104X
    [4X[28X-------------------------------------------[128X[104X
    [4X[28X a | [ 2 ]   [ 2 ]   [ 2 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X b | [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [ 3 ]   [128X[104X
    [4X[28X c | [ 4 ]   [ 4 ]   [ 5 ]   [ 4 ]   [ 5 ]   [128X[104X
    [4X[28XInitial state:    [ 1 ][128X[104X
    [4X[28XAccepting states: [ 3, 5 ][128X[104X
    [4X[25Xgap>[125X [27X [127X[104X
  [4X[32X[104X
  
  [1X9.3-7 NonSimpleAut[101X
  
  [33X[1;0Y[29X[2XNonSimpleAut[102X( [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe  automaton  accepting all rank encoded non-simple permutations
            with rank encoding [10Xk[110X.[133X
  
  [33X[0;0YWe find the language of all non-simple permutations of the set of all [22Xk[122X rank
  encoded permutations [22XΩ_k[122X using the above equation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:=NonSimpleAut(3);[127X[104X
    [4X[28X< deterministic automaton on 3 letters with 9 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(a);[127X[104X
    [4X[28X   |  1  2  3  4  5  6  7  8  9  [128X[104X
    [4X[28X--------------------------------[128X[104X
    [4X[28X a |  1  3  1  5  3  1  6  3  3  [128X[104X
    [4X[28X b |  3  3  3  3  9  9  3  9  3  [128X[104X
    [4X[28X c |  2  2  2  2  4  4  2  7  4  [128X[104X
    [4X[28XInitial state:   [ 8 ][128X[104X
    [4X[28XAccepting state: [ 1 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  
  [1X9.4 [33X[0;0YSimplicity[133X[101X
  
  [33X[0;0YThe set of simple permutations of a class is the complement set of the class
  with the non-simple permutations. We are working in the rank encoding and so
  in  language  terms our set of simple permutations [22XS_k[122X will be the subset of
  [22XΩ_k[122X[133X
  
  
  [24X[33X[0;6YE(S_k) = E(Ω_k∖ NS_k) = E(Ω_k) ∖ E(NS_k) = E(Ω_k) ∩ E(NS_k)^C[133X[124X
  
  [1X9.4-1 SimplePermAut[101X
  
  [33X[1;0Y[29X[2XSimplePermAut[102X( [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe  automaton accepting all rank encoded simple permutations with
            highest rank encoding [10Xk[110X.[133X
  
  [33X[0;0YWe  find  the  language  of all simple permutations of the set of all [22Xk[122X rank
  encoded permutations [22XΩ_k[122X using the above equation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSimplePermAut(3);[127X[104X
    [4X[28X< deterministic automaton on 3 letters with 8 states >[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28X   |  1  2  3  4  5  6  7  8  [128X[104X
    [4X[28X-----------------------------[128X[104X
    [4X[28X a |  2  2  1  1  7  2  1  6  [128X[104X
    [4X[28X b |  2  2  4  2  2  4  4  2  [128X[104X
    [4X[28X c |  2  2  8  5  2  5  5  2  [128X[104X
    [4X[28XInitial state:    [ 3 ][128X[104X
    [4X[28XAccepting states: [ 1, 3 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  
  [1X9.5 [33X[0;0YExceptionality[133X[101X
  
  [33X[0;0YA  permutation  is  said  to be exceptional if it is of one of the following
  forms,[133X
  
  
  [24X[33X[0;6Y2 4 6 ... (2m) 1 3 5 ... (2m-1)[133X[124X
  
  
  [24X[33X[0;6Y(2m-1) (2m-3) ... 1 (2m) (2m-2) ... 2[133X[124X
  
  
  [24X[33X[0;6Y(m+1) 1 (m+2) 2 (m+3) 3 ... (2m) m[133X[124X
  
  
  [24X[33X[0;6Ym (2m) (m-1) (2m-1) ... 1 (m+1)[133X[124X
  
  [33X[0;0Ywhere [22Xm ≥ 2[122X [7].[133X
  
  [1X9.5-1 IsExceptionalPerm[101X
  
  [33X[1;0Y[29X[2XIsExceptionalPerm[102X( [3Xperm[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[10XTrue[110X if [10Xperm[110X is exceptional, [10XFalse[110X otherwise.[133X
  
  [33X[0;0YThe  functions  checks  whether  [10Xperm[110X  is  one of the 4 types of exceptional
  permutations, that are described above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsExceptionalPerm([1,2,5,3,4]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsExceptionalPerm([1,1,3,1,1]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsExceptionalPerm([2,4,6,1,3,5]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsExceptionalPerm([2,3,4,1,1,1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [1X9.5-2 ExceptionalBoundedAutomaton[101X
  
  [33X[1;0Y[29X[2XExceptionalBoundedAutomaton[102X( [3Xk[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YThe  automaton  which  accepts  all  exceptional permutations with
            highest rank encoding [10Xk[110X.[133X
  
  [33X[0;0YThe  language of [10Xk[110X rank encoded exceptional permutations will be finite, and
  so it regular.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XExceptionalBoundedAutomaton(8); [127X[104X
    [4X[28X< deterministic automaton on 8 letters with 41 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(last,20);             [127X[104X
    [4X[28X[ 0, 2, 0, 2, 0, 4, 0, 4, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XExceptionalBoundedAutomaton(5);[127X[104X
    [4X[28X< deterministic automaton on 5 letters with 21 states >[128X[104X
    [4X[25Xgap>[125X [27XSpectrum(last);[127X[104X
    [4X[28X[ 0, 2, 0, 2, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
