Lehmer-Stanley-Baltic Rational Generating Functions for Enumerating CIRCULAR permutations

that don't go very far, Using a Zeilberger-style, Empirical-Yet-Rigorous , Dumb-Yet-Clever, Approach



                              By Shalosh B. Ekhad



        Let F[S](t) be the generating function, in t, for the number of

 CIRCULAR permutations of {1, ...n} such that pi[i]-i is always in S, where \
    things get wraped up.

          In this article we will derive, empirically-yet-rigorously,

Lehmer-Stanley-Baltic rational generating functions for all sets S of the fo\
    rm

                 {a,a+1, .., b} for -N<=a<-1<1<b<=N, where N=, 3

For the sake of Sloane, we will also provide the first 30 terms of the enume\
    rating sequence



                      ------------------------------------



                       If S=, {-3, -2, -1, 0, 1}, then :

                            15       14       13        12        11        10
generating function:, (-41 t   - 64 t   + 48 t   + 113 t   + 213 t   + 190 t

            9        8        7       6       5       4      3    2
     - 122 t  - 158 t  - 150 t  - 75 t  + 60 t  + 10 t  + 2 t  - t  - 2 t + 1)

       /   11    10      9      8      7      6      5      4      3
      /  (t   + t   - 2 t  - 2 t  - 4 t  - 2 t  + 6 t  + 2 t  + 2 t  - 3 t + 1)
     /

                                 Maple format:

(-41*t^15-64*t^14+48*t^13+113*t^12+213*t^11+190*t^10-122*t^9-158*t^8-150*t^7-75
*t^6+60*t^5+10*t^4+2*t^3-t^2-2*t+1)/(t^11+t^10-2*t^9-2*t^8-4*t^7-2*t^6+6*t^5+2*
t^4+2*t^3-3*t+1)


seq.: , [1, 2, 6, 24, 120, 265, 579, 1265, 2783, 6208, 13909, 31337, 70985,

    161545, 369024, 845825, 1944295, 4480285, 10345391, 23930320]

                      ------------------------------------



                      If S=, {-3, -2, -1, 0, 1, 2}, then :



seq.: , [1, 2, 6, 24, 120, 720, 1854, 4738, 12072, 30818, 79118, 204448, 528950,

    1370674, 3557408, 9244418, 24043990, 62573616, 162925614, 424377730]

                      ------------------------------------



                    If S=, {-3, -2, -1, 0, 1, 2, 3}, then :



seq.: , [1, 2, 6, 24, 120, 720, 5040, 14833, 43387, 126565, 369321, 1081313,

    3182225, 9411840, 27888139, 82819713, 246529859, 735516229, 2198971201,

    6586622953]

                      ------------------------------------



                         If S=, {-2, -1, 0, 1}, then :

                                  7      6      5       4      3
                             -10 t  - 6 t  - 3 t  + 13 t  + 2 t  - t + 1
       generating function:, -------------------------------------------
                                             4
                                            t  - 2 t + 1

                                 Maple format:

(-10*t^7-6*t^6-3*t^5+13*t^4+2*t^3-t+1)/(t^4-2*t+1)


seq.: , [1, 2, 6, 24, 44, 80, 144, 264, 484, 888, 1632, 3000, 5516, 10144,

    18656, 34312, 63108, 116072, 213488, 392664]

                      ------------------------------------



                        If S=, {-2, -1, 0, 1, 2}, then :

                            15       14       13        12        11        10
generating function:, (-41 t   - 64 t   + 48 t   + 113 t   + 213 t   + 190 t

            9        8        7       6       5       4      3    2
     - 122 t  - 158 t  - 150 t  - 75 t  + 60 t  + 10 t  + 2 t  - t  - 2 t + 1)

       /   11    10      9      8      7      6      5      4      3
      /  (t   + t   - 2 t  - 2 t  - 4 t  - 2 t  + 6 t  + 2 t  + 2 t  - 3 t + 1)
     /

                                 Maple format:

(-41*t^15-64*t^14+48*t^13+113*t^12+213*t^11+190*t^10-122*t^9-158*t^8-150*t^7-75
*t^6+60*t^5+10*t^4+2*t^3-t^2-2*t+1)/(t^11+t^10-2*t^9-2*t^8-4*t^7-2*t^6+6*t^5+2*
t^4+2*t^3-3*t+1)


seq.: , [1, 2, 6, 24, 120, 265, 579, 1265, 2783, 6208, 13909, 31337, 70985,

    161545, 369024, 845825, 1944295, 4480285, 10345391, 23930320]

                      ------------------------------------



                      If S=, {-2, -1, 0, 1, 2, 3}, then :



seq.: , [1, 2, 6, 24, 120, 720, 1854, 4738, 12072, 30818, 79118, 204448, 528950,

    1370674, 3557408, 9244418, 24043990, 62573616, 162925614, 424377730]

                      ------------------------------------



                           If S=, {-1, 0, 1}, then :

                                         5      4      3
                                     -3 t  - 2 t  + 3 t  - t + 1
               generating function:, ---------------------------
                                             3
                                            t  - 2 t + 1

                                 Maple format:

(-3*t^5-2*t^4+3*t^3-t+1)/(t^3-2*t+1)


seq.: , [1, 2, 6, 9, 13, 20, 31, 49, 78, 125, 201, 324, 523, 845, 1366, 2209,

    3573, 5780, 9351, 15129]

                      ------------------------------------



                          If S=, {-1, 0, 1, 2}, then :

                                  7      6      5       4      3
                             -10 t  - 6 t  - 3 t  + 13 t  + 2 t  - t + 1
       generating function:, -------------------------------------------
                                             4
                                            t  - 2 t + 1

                                 Maple format:

(-10*t^7-6*t^6-3*t^5+13*t^4+2*t^3-t+1)/(t^4-2*t+1)


seq.: , [1, 2, 6, 24, 44, 80, 144, 264, 484, 888, 1632, 3000, 5516, 10144,

    18656, 34312, 63108, 116072, 213488, 392664]

                      ------------------------------------



                        If S=, {-1, 0, 1, 2, 3}, then :

                            15       14       13        12        11        10
generating function:, (-41 t   - 64 t   + 48 t   + 113 t   + 213 t   + 190 t

            9        8        7       6       5       4      3    2
     - 122 t  - 158 t  - 150 t  - 75 t  + 60 t  + 10 t  + 2 t  - t  - 2 t + 1)

       /   11    10      9      8      7      6      5      4      3
      /  (t   + t   - 2 t  - 2 t  - 4 t  - 2 t  + 6 t  + 2 t  + 2 t  - 3 t + 1)
     /

                                 Maple format:

(-41*t^15-64*t^14+48*t^13+113*t^12+213*t^11+190*t^10-122*t^9-158*t^8-150*t^7-75
*t^6+60*t^5+10*t^4+2*t^3-t^2-2*t+1)/(t^11+t^10-2*t^9-2*t^8-4*t^7-2*t^6+6*t^5+2*
t^4+2*t^3-3*t+1)


seq.: , [1, 2, 6, 24, 120, 265, 579, 1265, 2783, 6208, 13909, 31337, 70985,

    161545, 369024, 845825, 1944295, 4480285, 10345391, 23930320]

     This ends this masterpiece, that took, 7264.861, seconds to generate.