The generating function of the number of reduced decompostions of , [1],  is ,

    1



                             and in Maple notation



1


                 The expectation is, 0, and the variance is, 0





The generating function of the number of reduced decompostions of , [2, 1],

     is , 1



                             and in Maple notation



1


                 The expectation is, 0, and the variance is, 0





The generating function of the number of reduced decompostions of , [3, 2, 1],

     is , 2 x



                             and in Maple notation



2*x


                 The expectation is, 1, and the variance is, 0





The generating function of the number of reduced decompostions of ,

                           2
    [4, 3, 2, 1],  is , 4 x  + 8 x + 4



                             and in Maple notation



4*x^2+8*x+4


                The expectation is, 1, and the variance is, 1/2





The generating function of the number of reduced decompostions of ,

                               3        2
    [5, 4, 3, 2, 1],  is , 32 x  + 192 x  + 288 x + 256



                             and in Maple notation



32*x^3+192*x^2+288*x+256


                The expectation is, 1, and the variance is, 3/4





The generating function of the number of reduced decompostions of ,

    [6, 5, 4, 3, 2, 1],  is ,

       6       5         4          3          2
    4 x  + 56 x  + 2152 x  + 16864 x  + 59780 x  + 113800 x + 100208



                             and in Maple notation



4*x^6+56*x^5+2152*x^4+16864*x^3+59780*x^2+113800*x+100208


                                                             11
                 The expectation is, 1, and the variance is, --
                                                             13





The generating function of the number of reduced decompostions of ,

                                       7          6            5             4
    [7, 6, 5, 4, 3, 2, 1],  is , 1456 x  + 67064 x  + 1236400 x  + 11741236 x

                 3              2
     + 64411616 x  + 215354704 x  + 423238880 x + 384691300



                             and in Maple notation



1456*x^7+67064*x^6+1236400*x^5+11741236*x^4+64411616*x^3+215354704*x^2+
423238880*x+384691300


                                                             17
                 The expectation is, 1, and the variance is, --
                                                             19





The generating function of the number of reduced decompostions of ,

                                          10           9             8
    [8, 7, 6, 5, 4, 3, 2, 1],  is , 1760 x   + 262160 x  + 16221264 x

                  7               6                5                 4
     + 459797952 x  + 7673875968 x  + 82084933968 x  + 589039596960 x

                      3                  2
     + 2874227383136 x  + 9331832099808 x  + 18506472293632 x + 17216989222352



                             and in Maple notation



1760*x^10+262160*x^9+16221264*x^8+459797952*x^7+7673875968*x^6+82084933968*x^5+
589039596960*x^4+2874227383136*x^3+9331832099808*x^2+18506472293632*x+
17216989222352


                                                             12
                 The expectation is, 1, and the variance is, --
                                                             13





The generating function of the number of reduced decompostions of ,

                                              13            12              11
    [9, 8, 7, 6, 5, 4, 3, 2, 1],  is , 10960 x   + 5034544 x   + 698402288 x

                    10                  9                   8
     + 42817670128 x   + 1525315547632 x  + 35652331972512 x

                        7                     6                      5
     + 581895590060208 x  + 6862185937112976 x  + 59435324031326192 x

                           4                        3                        2
     + 378559369042744528 x  + 1743764520961564288 x  + 5555432536259193248 x

     + 11059248497359308336 x + 10454445445908540480



                             and in Maple notation



10960*x^13+5034544*x^12+698402288*x^11+42817670128*x^10+1525315547632*x^9+
35652331972512*x^8+581895590060208*x^7+6862185937112976*x^6+59435324031326192*x
^5+378559369042744528*x^4+1743764520961564288*x^3+5555432536259193248*x^2+
11059248497359308336*x+10454445445908540480


                                                             16
                 The expectation is, 1, and the variance is, --
                                                             17





              To sum up, the first , 9, generating functions are



[1, 1, 2*x, 4*x^2+8*x+4, 32*x^3+192*x^2+288*x+256, 4*x^6+56*x^5+2152*x^4+16864*
x^3+59780*x^2+113800*x+100208, 1456*x^7+67064*x^6+1236400*x^5+11741236*x^4+
64411616*x^3+215354704*x^2+423238880*x+384691300, 1760*x^10+262160*x^9+16221264
*x^8+459797952*x^7+7673875968*x^6+82084933968*x^5+589039596960*x^4+
2874227383136*x^3+9331832099808*x^2+18506472293632*x+17216989222352, 10960*x^13
+5034544*x^12+698402288*x^11+42817670128*x^10+1525315547632*x^9+35652331972512*
x^8+581895590060208*x^7+6862185937112976*x^6+59435324031326192*x^5+
378559369042744528*x^4+1743764520961564288*x^3+5555432536259193248*x^2+
11059248497359308336*x+10454445445908540480]


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