Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 1
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 1
[4] generating graphics... => 2
[3,1] generating graphics... => 1
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 4
[4,1] generating graphics... => 2
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 1
[2,2,1] generating graphics... => 1
[2,1,1,1] generating graphics... => 1
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 11
[5,1] generating graphics... => 4
[4,2] generating graphics... => 5
[4,1,1] generating graphics... => 2
[3,3] generating graphics... => 2
[3,2,1] generating graphics... => 2
[3,1,1,1] generating graphics... => 1
[2,2,2] generating graphics... => 1
[2,2,1,1] generating graphics... => 1
[2,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 33
[6,1] generating graphics... => 11
[5,2] generating graphics... => 12
[5,1,1] generating graphics... => 4
[4,3] generating graphics... => 10
[4,2,1] generating graphics... => 5
[4,1,1,1] generating graphics... => 2
[3,3,1] generating graphics... => 2
[3,2,2] generating graphics... => 3
[3,2,1,1] generating graphics... => 2
[3,1,1,1,1] generating graphics... => 1
[2,2,2,1] generating graphics... => 1
[2,2,1,1,1] generating graphics... => 1
[2,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 116
[7,1] generating graphics... => 33
[6,2] generating graphics... => 37
[6,1,1] generating graphics... => 11
[5,3] generating graphics... => 27
[5,2,1] generating graphics... => 12
[5,1,1,1] generating graphics... => 4
[4,4] generating graphics... => 19
[4,3,1] generating graphics... => 10
[4,2,2] generating graphics... => 9
[4,2,1,1] generating graphics... => 5
[4,1,1,1,1] generating graphics... => 2
[3,3,2] generating graphics... => 5
[3,3,1,1] generating graphics... => 2
[3,2,2,1] generating graphics... => 3
[3,2,1,1,1] generating graphics... => 2
[3,1,1,1,1,1] generating graphics... => 1
[2,2,2,2] generating graphics... => 1
[2,2,2,1,1] generating graphics... => 1
[2,2,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 435
[8,1] generating graphics... => 116
[7,2] generating graphics... => 123
[7,1,1] generating graphics... => 33
[6,3] generating graphics... => 97
[6,2,1] generating graphics... => 37
[6,1,1,1] generating graphics... => 11
[5,4] generating graphics... => 99
[5,3,1] generating graphics... => 27
[5,2,2] generating graphics... => 25
[5,2,1,1] generating graphics... => 12
[5,1,1,1,1] generating graphics... => 4
[4,4,1] generating graphics... => 19
[4,3,2] generating graphics... => 28
[4,3,1,1] generating graphics... => 10
[4,2,2,1] generating graphics... => 9
[4,2,1,1,1] generating graphics... => 5
[4,1,1,1,1,1] generating graphics... => 2
[3,3,3] generating graphics... => 5
[3,3,2,1] generating graphics... => 5
[3,3,1,1,1] generating graphics... => 2
[3,2,2,2] generating graphics... => 4
[3,2,2,1,1] generating graphics... => 3
[3,2,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,1] generating graphics... => 1
[2,2,2,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 1832
[9,1] generating graphics... => 435
[8,2] generating graphics... => 474
[8,1,1] generating graphics... => 116
[7,3] generating graphics... => 351
[7,2,1] generating graphics... => 123
[7,1,1,1] generating graphics... => 33
[6,4] generating graphics... => 384
[6,3,1] generating graphics... => 97
[6,2,2] generating graphics... => 85
[6,2,1,1] generating graphics... => 37
[6,1,1,1,1] generating graphics... => 11
[5,5] generating graphics... => 188
[5,4,1] generating graphics... => 99
[5,3,2] generating graphics... => 89
[5,3,1,1] generating graphics... => 27
[5,2,2,1] generating graphics... => 25
[5,2,1,1,1] generating graphics... => 12
[5,1,1,1,1,1] generating graphics... => 4
[4,4,2] generating graphics... => 61
[4,4,1,1] generating graphics... => 19
[4,3,3] generating graphics... => 42
[4,3,2,1] generating graphics... => 28
[4,3,1,1,1] generating graphics... => 10
[4,2,2,2] generating graphics... => 14
[4,2,2,1,1] generating graphics... => 9
[4,2,1,1,1,1] generating graphics... => 5
[4,1,1,1,1,1,1] generating graphics... => 2
[3,3,3,1] generating graphics... => 5
[3,3,2,2] generating graphics... => 9
[3,3,2,1,1] generating graphics... => 5
[3,3,1,1,1,1] generating graphics... => 2
[3,2,2,2,1] generating graphics... => 4
[3,2,2,1,1,1] generating graphics... => 3
[3,2,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2] generating graphics... => 1
[2,2,2,2,1,1] generating graphics... => 1
[2,2,2,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 8167
[10,1] generating graphics... => 1832
[9,2] generating graphics... => 1907
[9,1,1] generating graphics... => 435
[8,3] generating graphics... => 1470
[8,2,1] generating graphics... => 474
[8,1,1,1] generating graphics... => 116
[7,4] generating graphics... => 1551
[7,3,1] generating graphics... => 351
[7,2,2] generating graphics... => 308
[7,2,1,1] generating graphics... => 123
[7,1,1,1,1] generating graphics... => 33
[6,5] generating graphics... => 1407
[6,4,1] generating graphics... => 384
[6,3,2] generating graphics... => 341
[6,3,1,1] generating graphics... => 97
[6,2,2,1] generating graphics... => 85
[6,2,1,1,1] generating graphics... => 37
[6,1,1,1,1,1] generating graphics... => 11
[5,5,1] generating graphics... => 188
[5,4,2] generating graphics... => 349
[5,4,1,1] generating graphics... => 99
[5,3,3] generating graphics... => 145
[5,3,2,1] generating graphics... => 89
[5,3,1,1,1] generating graphics... => 27
[5,2,2,2] generating graphics... => 44
[5,2,2,1,1] generating graphics... => 25
[5,2,1,1,1,1] generating graphics... => 12
[5,1,1,1,1,1,1] generating graphics... => 4
[4,4,3] generating graphics... => 159
[4,4,2,1] generating graphics... => 61
[4,4,1,1,1] generating graphics... => 19
[4,3,3,1] generating graphics... => 42
[4,3,2,2] generating graphics... => 56
[4,3,2,1,1] generating graphics... => 28
[4,3,1,1,1,1] generating graphics... => 10
[4,2,2,2,1] generating graphics... => 14
[4,2,2,1,1,1] generating graphics... => 9
[4,2,1,1,1,1,1] generating graphics... => 5
[4,1,1,1,1,1,1,1] generating graphics... => 2
[3,3,3,2] generating graphics... => 14
[3,3,3,1,1] generating graphics... => 5
[3,3,2,2,1] generating graphics... => 9
[3,3,2,1,1,1] generating graphics... => 5
[3,3,1,1,1,1,1] generating graphics... => 2
[3,2,2,2,2] generating graphics... => 5
[3,2,2,2,1,1] generating graphics... => 4
[3,2,2,1,1,1,1] generating graphics... => 3
[3,2,1,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2,1] generating graphics... => 1
[2,2,2,2,1,1,1] generating graphics... => 1
[2,2,2,1,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 39700
[11,1] generating graphics... => 8167
[10,2] generating graphics... => 8593
[10,1,1] generating graphics... => 1832
[9,3] generating graphics... => 6314
[9,2,1] generating graphics... => 1907
[9,1,1,1] generating graphics... => 435
[8,4] generating graphics... => 7084
[8,3,1] generating graphics... => 1470
[8,2,2] generating graphics... => 1285
[8,2,1,1] generating graphics... => 474
[8,1,1,1,1] generating graphics... => 116
[7,5] generating graphics... => 6009
[7,4,1] generating graphics... => 1551
[7,3,2] generating graphics... => 1329
[7,3,1,1] generating graphics... => 351
[7,2,2,1] generating graphics... => 308
[7,2,1,1,1] generating graphics... => 123
[7,1,1,1,1,1] generating graphics... => 33
[6,6] generating graphics... => 3533
[6,5,1] generating graphics... => 1407
[6,4,2] generating graphics... => 1500
[6,4,1,1] generating graphics... => 384
[6,3,3] generating graphics... => 626
[6,3,2,1] generating graphics... => 341
[6,3,1,1,1] generating graphics... => 97
[6,2,2,2] generating graphics... => 163
[6,2,2,1,1] generating graphics... => 85
[6,2,1,1,1,1] generating graphics... => 37
[6,1,1,1,1,1,1] generating graphics... => 11
[5,5,2] generating graphics... => 740
[5,5,1,1] generating graphics... => 188
[5,4,3] generating graphics... => 982
[5,4,2,1] generating graphics... => 349
[5,4,1,1,1] generating graphics... => 99
[5,3,3,1] generating graphics... => 145
[5,3,2,2] generating graphics... => 203
[5,3,2,1,1] generating graphics... => 89
[5,3,1,1,1,1] generating graphics... => 27
[5,2,2,2,1] generating graphics... => 44
[5,2,2,1,1,1] generating graphics... => 25
[5,2,1,1,1,1,1] generating graphics... => 12
[5,1,1,1,1,1,1,1] generating graphics... => 4
[4,4,4] generating graphics... => 296
[4,4,3,1] generating graphics... => 159
[4,4,2,2] generating graphics... => 137
[4,4,2,1,1] generating graphics... => 61
[4,4,1,1,1,1] generating graphics... => 19
[4,3,3,2] generating graphics... => 126
[4,3,3,1,1] generating graphics... => 42
[4,3,2,2,1] generating graphics... => 56
[4,3,2,1,1,1] generating graphics... => 28
[4,3,1,1,1,1,1] generating graphics... => 10
[4,2,2,2,2] generating graphics... => 20
[4,2,2,2,1,1] generating graphics... => 14
[4,2,2,1,1,1,1] generating graphics... => 9
[4,2,1,1,1,1,1,1] generating graphics... => 5
[4,1,1,1,1,1,1,1,1] generating graphics... => 2
[3,3,3,3] generating graphics... => 14
[3,3,3,2,1] generating graphics... => 14
[3,3,3,1,1,1] generating graphics... => 5
[3,3,2,2,2] generating graphics... => 14
[3,3,2,2,1,1] generating graphics... => 9
[3,3,2,1,1,1,1] generating graphics... => 5
[3,3,1,1,1,1,1,1] generating graphics... => 2
[3,2,2,2,2,1] generating graphics... => 5
[3,2,2,2,1,1,1] generating graphics... => 4
[3,2,2,1,1,1,1,1] generating graphics... => 3
[3,2,1,1,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2,2] generating graphics... => 1
[2,2,2,2,2,1,1] generating graphics... => 1
[2,2,2,2,1,1,1,1] generating graphics... => 1
[2,2,2,1,1,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[5,4,3,1] generating graphics... => 982
[5,4,2,2] generating graphics... => 855
[5,4,2,1,1] generating graphics... => 349
[5,3,3,2] generating graphics... => 502
[5,3,3,1,1] generating graphics... => 145
[5,3,2,2,1] generating graphics... => 203
[4,4,3,2] generating graphics... => 518
[4,4,3,1,1] generating graphics... => 159
[4,4,2,2,1] generating graphics... => 137
[4,3,3,2,1] generating graphics... => 126
[5,4,3,2] generating graphics... => 3516
[5,4,3,1,1] generating graphics... => 982
[5,4,2,2,1] generating graphics... => 855
[5,3,3,2,1] generating graphics... => 502
[4,4,3,2,1] generating graphics... => 518
[5,4,3,2,1] generating graphics... => 3516
click to show generating function       
Description
The number of ways to refine the partition into singletons.
For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$.
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.
Code
def statistic(la):
    P = posets.IntegerPartitions(la.size())
    H = P.hasse_diagram()
    e = H.vertices()[0]
    f = tuple(la)
    return len(H.all_simple_paths([f], [e], trivial=True))

Created
Jul 19, 2017 at 15:58 by Martin Rubey
Updated
Jan 17, 2018 at 23:29 by Martin Rubey