#neworder.ranks

This file contains the results of calculations of the rank statistics of
elements in the universal k-Bruhat order on S_n, for small values of n.   
This was originally computed  at MSRI in May, 1997 with the file 
neworder.maple, and extended to n=12 recently using the file neworder.c.

Frank Sottile
2 January, 1998
Toronto

##################################################

        
 n|                      Ranks           
  | 0  1    2     3      4      5      6       7       8       9        10     
--|---------------------------------------------------------------------------
 1| 1
 2| 1  1
 3| 1  3    2
 4| 1  6   10     6      1
 5| 1 10   30    40     27     10       2
 6| 1 15   70   155    195    156      86      33       8        1
 7| 1 21  140   455    875   1120    1038     735     406      175       58 
 8| 1 28  252  1120   2996   5432    7252    7562    6398     4492     2652
 9| 1 36  420  2436   8568  20580   36624   51228   58748    56868    47412 
10| 1 45  660  4830  21504  65562  148344  264090  386220   478886   515202
11| 1 55  990  8910  48840 183414  510972 1118766 2010096  3061652  4051544
12| 1 66 1430 15510 102465 463452 1552848 4080780 8780277 15984320 25251666 
--|-------------------------------------------------------------------------
  | 0  1    2     3     4      5       6       7       8        9       10    
 
7 |       14        2
8 |     1324      556      192       52       10        1
9 |    34530    22170    12600     6332     2796     1071      348       92 
10|   489243   415380   318265   221454   140447    81291    42900    20570
11|  4746159  4993472  4771767  4177624  3372809  2523169  1754830  1136940
12| 35304260 44361284 50714994 53267918 51805820 46942830 39826048 31756582 
--|-------------------------------------------------------------------------
  |     11      12        13       14      15        16       17       18   

9 |       18        2
10|     8899     3432     1157     330      75      12      1
11|   686741   386518   202301   98098   43813   17866    6567   2134    594 
12| 23868858 16946074 11379522 7231926 4349046 2472525 1326534 669756 316942
--|--------------------------------------------------------------------------
  |     19       20       21      22      23       24       25     26    27  

11|    134    22     2
12| 139776 56992 21244 7122 2095 520 102  14   1 
--|-----------------------------------------------------------
  |    28     29    30   31   32  33  34  35  36 


      |  Symmetric groups
ranks | 1 2 3  4  5   6    7    8     9     10     11        12|
      |                                                        |   
  0   | 1 1 1  1  1   1    1    1     1      1       1        1|
  1   |   1 3  6 10  15   21   28    36     45	    55       66|  n\choose 2
  2   |     2 10 30  70  140  252   420    660     990     1430|  2*{n+4\choose 4}
  3   |        6 40 155  455 1120  2436   4830    8910    15510| 3      /\
  4   |        1 27 195  875 2996  8568  21504   48840   102465| 4      ||
  5   |          10 156 1120 5432 20580  65562  183414   463452| 5   noted by
  6   |           2  86 1038 7252 36624 148344  510972  1552848| 6     Einar
  7   |              33  735 7562 51228 264090 1118766  4080780| 7  Steingrimsson
  8   |               8  406 6398 58748 386220 2010096  8780277| 8
  9   |               1  175 4492 56868 478886 3061652 15984320| 9
 10   |                   58 2652 47412 515202 4051544 25251666| 10
 11   |                   14 1324 34530 489243 4746159 35304260| 11
 12   |                    2  556 22170 415380 4993472 44361284| 12
 13   |                       192 12600 318265 4771767 50714994| 13
 14   |                        52  6332 221454 4177624 53267918| 14
 15   |                        10  2796 140447 3372809 51805820| 15
 16   |                         1  1071  81291 2523169 46942830| 16
 17   |                             348  42900 1754830 39826048| 17
 18   |                              92  20570 1136940 31756582| 18
                                     18   8899  686741 23868858| 19
                                      2   3432  386518 16946074| 20
                                          1157  202301 11379522| 21
                                           330   98098  7231926| 22 
                                            75   43813  4349046| 23
                                            12   17866  2472525| 24
                                             1    6567  1326534| 25
                                                  2134   669756| 26
                                                   594   316942| 27
                                                   134   139776| 28
                                                    22    56992| 28
                                                     2    21244| 30
                                                           7122| 31
                                                           2095| 32
                                                            520| 33
                                                            102| 34
                                                             14| 35
                                                              1| 36