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Statistic identifier: St000913
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Collection: Integer partitions
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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]].
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References:
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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))
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Statistic values:
[] => 1
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 1
[1,1,1] => 1
[4] => 2
[3,1] => 1
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 4
[4,1] => 2
[3,2] => 2
[3,1,1] => 1
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 11
[5,1] => 4
[4,2] => 5
[4,1,1] => 2
[3,3] => 2
[3,2,1] => 2
[3,1,1,1] => 1
[2,2,2] => 1
[2,2,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 1
[7] => 33
[6,1] => 11
[5,2] => 12
[5,1,1] => 4
[4,3] => 10
[4,2,1] => 5
[4,1,1,1] => 2
[3,3,1] => 2
[3,2,2] => 3
[3,2,1,1] => 2
[3,1,1,1,1] => 1
[2,2,2,1] => 1
[2,2,1,1,1] => 1
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 1
[8] => 116
[7,1] => 33
[6,2] => 37
[6,1,1] => 11
[5,3] => 27
[5,2,1] => 12
[5,1,1,1] => 4
[4,4] => 19
[4,3,1] => 10
[4,2,2] => 9
[4,2,1,1] => 5
[4,1,1,1,1] => 2
[3,3,2] => 5
[3,3,1,1] => 2
[3,2,2,1] => 3
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 1
[2,2,2,2] => 1
[2,2,2,1,1] => 1
[2,2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 1
[9] => 435
[8,1] => 116
[7,2] => 123
[7,1,1] => 33
[6,3] => 97
[6,2,1] => 37
[6,1,1,1] => 11
[5,4] => 99
[5,3,1] => 27
[5,2,2] => 25
[5,2,1,1] => 12
[5,1,1,1,1] => 4
[4,4,1] => 19
[4,3,2] => 28
[4,3,1,1] => 10
[4,2,2,1] => 9
[4,2,1,1,1] => 5
[4,1,1,1,1,1] => 2
[3,3,3] => 5
[3,3,2,1] => 5
[3,3,1,1,1] => 2
[3,2,2,2] => 4
[3,2,2,1,1] => 3
[3,2,1,1,1,1] => 2
[3,1,1,1,1,1,1] => 1
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1832
[9,1] => 435
[8,2] => 474
[8,1,1] => 116
[7,3] => 351
[7,2,1] => 123
[7,1,1,1] => 33
[6,4] => 384
[6,3,1] => 97
[6,2,2] => 85
[6,2,1,1] => 37
[6,1,1,1,1] => 11
[5,5] => 188
[5,4,1] => 99
[5,3,2] => 89
[5,3,1,1] => 27
[5,2,2,1] => 25
[5,2,1,1,1] => 12
[5,1,1,1,1,1] => 4
[4,4,2] => 61
[4,4,1,1] => 19
[4,3,3] => 42
[4,3,2,1] => 28
[4,3,1,1,1] => 10
[4,2,2,2] => 14
[4,2,2,1,1] => 9
[4,2,1,1,1,1] => 5
[4,1,1,1,1,1,1] => 2
[3,3,3,1] => 5
[3,3,2,2] => 9
[3,3,2,1,1] => 5
[3,3,1,1,1,1] => 2
[3,2,2,2,1] => 4
[3,2,2,1,1,1] => 3
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 1
[2,2,2,1,1,1,1] => 1
[2,2,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 8167
[10,1] => 1832
[9,2] => 1907
[9,1,1] => 435
[8,3] => 1470
[8,2,1] => 474
[8,1,1,1] => 116
[7,4] => 1551
[7,3,1] => 351
[7,2,2] => 308
[7,2,1,1] => 123
[7,1,1,1,1] => 33
[6,5] => 1407
[6,4,1] => 384
[6,3,2] => 341
[6,3,1,1] => 97
[6,2,2,1] => 85
[6,2,1,1,1] => 37
[6,1,1,1,1,1] => 11
[5,5,1] => 188
[5,4,2] => 349
[5,4,1,1] => 99
[5,3,3] => 145
[5,3,2,1] => 89
[5,3,1,1,1] => 27
[5,2,2,2] => 44
[5,2,2,1,1] => 25
[5,2,1,1,1,1] => 12
[5,1,1,1,1,1,1] => 4
[4,4,3] => 159
[4,4,2,1] => 61
[4,4,1,1,1] => 19
[4,3,3,1] => 42
[4,3,2,2] => 56
[4,3,2,1,1] => 28
[4,3,1,1,1,1] => 10
[4,2,2,2,1] => 14
[4,2,2,1,1,1] => 9
[4,2,1,1,1,1,1] => 5
[4,1,1,1,1,1,1,1] => 2
[3,3,3,2] => 14
[3,3,3,1,1] => 5
[3,3,2,2,1] => 9
[3,3,2,1,1,1] => 5
[3,3,1,1,1,1,1] => 2
[3,2,2,2,2] => 5
[3,2,2,2,1,1] => 4
[3,2,2,1,1,1,1] => 3
[3,2,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => 1
[2,2,2,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 39700
[11,1] => 8167
[10,2] => 8593
[10,1,1] => 1832
[9,3] => 6314
[9,2,1] => 1907
[9,1,1,1] => 435
[8,4] => 7084
[8,3,1] => 1470
[8,2,2] => 1285
[8,2,1,1] => 474
[8,1,1,1,1] => 116
[7,5] => 6009
[7,4,1] => 1551
[7,3,2] => 1329
[7,3,1,1] => 351
[7,2,2,1] => 308
[7,2,1,1,1] => 123
[7,1,1,1,1,1] => 33
[6,6] => 3533
[6,5,1] => 1407
[6,4,2] => 1500
[6,4,1,1] => 384
[6,3,3] => 626
[6,3,2,1] => 341
[6,3,1,1,1] => 97
[6,2,2,2] => 163
[6,2,2,1,1] => 85
[6,2,1,1,1,1] => 37
[6,1,1,1,1,1,1] => 11
[5,5,2] => 740
[5,5,1,1] => 188
[5,4,3] => 982
[5,4,2,1] => 349
[5,4,1,1,1] => 99
[5,3,3,1] => 145
[5,3,2,2] => 203
[5,3,2,1,1] => 89
[5,3,1,1,1,1] => 27
[5,2,2,2,1] => 44
[5,2,2,1,1,1] => 25
[5,2,1,1,1,1,1] => 12
[5,1,1,1,1,1,1,1] => 4
[4,4,4] => 296
[4,4,3,1] => 159
[4,4,2,2] => 137
[4,4,2,1,1] => 61
[4,4,1,1,1,1] => 19
[4,3,3,2] => 126
[4,3,3,1,1] => 42
[4,3,2,2,1] => 56
[4,3,2,1,1,1] => 28
[4,3,1,1,1,1,1] => 10
[4,2,2,2,2] => 20
[4,2,2,2,1,1] => 14
[4,2,2,1,1,1,1] => 9
[4,2,1,1,1,1,1,1] => 5
[4,1,1,1,1,1,1,1,1] => 2
[3,3,3,3] => 14
[3,3,3,2,1] => 14
[3,3,3,1,1,1] => 5
[3,3,2,2,2] => 14
[3,3,2,2,1,1] => 9
[3,3,2,1,1,1,1] => 5
[3,3,1,1,1,1,1,1] => 2
[3,2,2,2,2,1] => 5
[3,2,2,2,1,1,1] => 4
[3,2,2,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => 1
[2,2,2,2,2,1,1] => 1
[2,2,2,2,1,1,1,1] => 1
[2,2,2,1,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
[5,4,3,1] => 982
[5,4,2,2] => 855
[5,4,2,1,1] => 349
[5,3,3,2] => 502
[5,3,3,1,1] => 145
[5,3,2,2,1] => 203
[4,4,3,2] => 518
[4,4,3,1,1] => 159
[4,4,2,2,1] => 137
[4,3,3,2,1] => 126
[5,4,3,2] => 3516
[5,4,3,1,1] => 982
[5,4,2,2,1] => 855
[5,3,3,2,1] => 502
[4,4,3,2,1] => 518
[5,4,3,2,1] => 3516
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Created: Jul 19, 2017 at 15:58 by Martin Rubey
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Last Updated: Jan 17, 2018 at 23:29 by Martin Rubey