- FindStat

Identifier

St000913: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>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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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