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Statistic identifier: St000284
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
-----------------------------------------------------------------------------
References: [1] [[wikipedia:Plancherel_measure]]
-----------------------------------------------------------------------------
Code:
def statistic(L):
return L.standard_tableaux().cardinality()^2
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 4
[1,1,1] => 1
[4] => 1
[3,1] => 9
[2,2] => 4
[2,1,1] => 9
[1,1,1,1] => 1
[5] => 1
[4,1] => 16
[3,2] => 25
[3,1,1] => 36
[2,2,1] => 25
[2,1,1,1] => 16
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 25
[4,2] => 81
[4,1,1] => 100
[3,3] => 25
[3,2,1] => 256
[3,1,1,1] => 100
[2,2,2] => 25
[2,2,1,1] => 81
[2,1,1,1,1] => 25
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 36
[5,2] => 196
[5,1,1] => 225
[4,3] => 196
[4,2,1] => 1225
[4,1,1,1] => 400
[3,3,1] => 441
[3,2,2] => 441
[3,2,1,1] => 1225
[3,1,1,1,1] => 225
[2,2,2,1] => 196
[2,2,1,1,1] => 196
[2,1,1,1,1,1] => 36
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 49
[6,2] => 400
[6,1,1] => 441
[5,3] => 784
[5,2,1] => 4096
[5,1,1,1] => 1225
[4,4] => 196
[4,3,1] => 4900
[4,2,2] => 3136
[4,2,1,1] => 8100
[4,1,1,1,1] => 1225
[3,3,2] => 1764
[3,3,1,1] => 3136
[3,2,2,1] => 4900
[3,2,1,1,1] => 4096
[3,1,1,1,1,1] => 441
[2,2,2,2] => 196
[2,2,2,1,1] => 784
[2,2,1,1,1,1] => 400
[2,1,1,1,1,1,1] => 49
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 64
[7,2] => 729
[7,1,1] => 784
[6,3] => 2304
[6,2,1] => 11025
[6,1,1,1] => 3136
[5,4] => 1764
[5,3,1] => 26244
[5,2,2] => 14400
[5,2,1,1] => 35721
[5,1,1,1,1] => 4900
[4,4,1] => 7056
[4,3,2] => 28224
[4,3,1,1] => 46656
[4,2,2,1] => 46656
[4,2,1,1,1] => 35721
[4,1,1,1,1,1] => 3136
[3,3,3] => 1764
[3,3,2,1] => 28224
[3,3,1,1,1] => 14400
[3,2,2,2] => 7056
[3,2,2,1,1] => 26244
[3,2,1,1,1,1] => 11025
[3,1,1,1,1,1,1] => 784
[2,2,2,2,1] => 1764
[2,2,2,1,1,1] => 2304
[2,2,1,1,1,1,1] => 729
[2,1,1,1,1,1,1,1] => 64
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 81
[8,2] => 1225
[8,1,1] => 1296
[7,3] => 5625
[7,2,1] => 25600
[7,1,1,1] => 7056
[6,4] => 8100
[6,3,1] => 99225
[6,2,2] => 50625
[6,2,1,1] => 122500
[6,1,1,1,1] => 15876
[5,5] => 1764
[5,4,1] => 82944
[5,3,2] => 202500
[5,3,1,1] => 321489
[5,2,2,1] => 275625
[5,2,1,1,1] => 200704
[5,1,1,1,1,1] => 15876
[4,4,2] => 63504
[4,4,1,1] => 90000
[4,3,3] => 44100
[4,3,2,1] => 589824
[4,3,1,1,1] => 275625
[4,2,2,2] => 90000
[4,2,2,1,1] => 321489
[4,2,1,1,1,1] => 122500
[4,1,1,1,1,1,1] => 7056
[3,3,3,1] => 44100
[3,3,2,2] => 63504
[3,3,2,1,1] => 202500
[3,3,1,1,1,1] => 50625
[3,2,2,2,1] => 82944
[3,2,2,1,1,1] => 99225
[3,2,1,1,1,1,1] => 25600
[3,1,1,1,1,1,1,1] => 1296
[2,2,2,2,2] => 1764
[2,2,2,2,1,1] => 8100
[2,2,2,1,1,1,1] => 5625
[2,2,1,1,1,1,1,1] => 1225
[2,1,1,1,1,1,1,1,1] => 81
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 1
[10,1] => 100
[9,2] => 1936
[9,1,1] => 2025
[8,3] => 12100
[8,2,1] => 53361
[8,1,1,1] => 14400
[7,4] => 27225
[7,3,1] => 302500
[7,2,2] => 148225
[7,2,1,1] => 352836
[7,1,1,1,1] => 44100
[6,5] => 17424
[6,4,1] => 480249
[6,3,2] => 980100
[6,3,1,1] => 1517824
[6,2,2,1] => 1210000
[6,2,1,1,1] => 853776
[6,1,1,1,1,1] => 63504
[5,5,1] => 108900
[5,4,2] => 980100
[5,4,1,1] => 1334025
[5,3,3] => 435600
[5,3,2,1] => 5336100
[5,3,1,1,1] => 2371600
[5,2,2,2] => 680625
[5,2,2,1,1] => 2371600
[5,2,1,1,1,1] => 853776
[5,1,1,1,1,1,1] => 44100
[4,4,3] => 213444
[4,4,2,1] => 1742400
[4,4,1,1,1] => 680625
[4,3,3,1] => 1411344
[4,3,2,2] => 1742400
[4,3,2,1,1] => 5336100
[4,3,1,1,1,1] => 1210000
[4,2,2,2,1] => 1334025
[4,2,2,1,1,1] => 1517824
[4,2,1,1,1,1,1] => 352836
[4,1,1,1,1,1,1,1] => 14400
[3,3,3,2] => 213444
[3,3,3,1,1] => 435600
[3,3,2,2,1] => 980100
[3,3,2,1,1,1] => 980100
[3,3,1,1,1,1,1] => 148225
[3,2,2,2,2] => 108900
[3,2,2,2,1,1] => 480249
[3,2,2,1,1,1,1] => 302500
[3,2,1,1,1,1,1,1] => 53361
[3,1,1,1,1,1,1,1,1] => 2025
[2,2,2,2,2,1] => 17424
[2,2,2,2,1,1,1] => 27225
[2,2,2,1,1,1,1,1] => 12100
[2,2,1,1,1,1,1,1,1] => 1936
[2,1,1,1,1,1,1,1,1,1] => 100
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 1
[11,1] => 121
[10,2] => 2916
[10,1,1] => 3025
[9,3] => 23716
[9,2,1] => 102400
[9,1,1,1] => 27225
[8,4] => 75625
[8,3,1] => 793881
[8,2,2] => 379456
[8,2,1,1] => 893025
[8,1,1,1,1] => 108900
[7,5] => 88209
[7,4,1] => 1982464
[7,3,2] => 3705625
[7,3,1,1] => 5645376
[7,2,2,1] => 4322241
[7,2,1,1,1] => 2985984
[7,1,1,1,1,1] => 213444
[6,6] => 17424
[6,5,1] => 1334025
[6,4,2] => 7144929
[6,4,1,1] => 9486400
[6,3,3] => 2722500
[6,3,2,1] => 31719424
[6,3,1,1,1] => 13660416
[6,2,2,2] => 3705625
[6,2,2,1,1] => 12702096
[6,2,1,1,1,1] => 4410000
[6,1,1,1,1,1,1] => 213444
[5,5,2] => 1742400
[5,5,1,1] => 2205225
[5,4,3] => 4460544
[5,4,2,1] => 33350625
[5,4,1,1,1] => 12390400
[5,3,3,1] => 17288964
[5,3,2,2] => 19847025
[5,3,2,1,1] => 59290000
[5,3,1,1,1,1] => 12702096
[5,2,2,2,1] => 12390400
[5,2,2,1,1,1] => 13660416
[5,2,1,1,1,1,1] => 2985984
[5,1,1,1,1,1,1,1] => 108900
[4,4,4] => 213444
[4,4,3,1] => 8820900
[4,4,2,2] => 6969600
[4,4,2,1,1] => 19847025
[4,4,1,1,1,1] => 3705625
[4,3,3,2] => 8820900
[4,3,3,1,1] => 17288964
[4,3,2,2,1] => 33350625
[4,3,2,1,1,1] => 31719424
[4,3,1,1,1,1,1] => 4322241
[4,2,2,2,2] => 2205225
[4,2,2,2,1,1] => 9486400
[4,2,2,1,1,1,1] => 5645376
[4,2,1,1,1,1,1,1] => 893025
[4,1,1,1,1,1,1,1,1] => 27225
[3,3,3,3] => 213444
[3,3,3,2,1] => 4460544
[3,3,3,1,1,1] => 2722500
[3,3,2,2,2] => 1742400
[3,3,2,2,1,1] => 7144929
[3,3,2,1,1,1,1] => 3705625
[3,3,1,1,1,1,1,1] => 379456
[3,2,2,2,2,1] => 1334025
[3,2,2,2,1,1,1] => 1982464
[3,2,2,1,1,1,1,1] => 793881
[3,2,1,1,1,1,1,1,1] => 102400
[3,1,1,1,1,1,1,1,1,1] => 3025
[2,2,2,2,2,2] => 17424
[2,2,2,2,2,1,1] => 88209
[2,2,2,2,1,1,1,1] => 75625
[2,2,2,1,1,1,1,1,1] => 23716
[2,2,1,1,1,1,1,1,1,1] => 2916
[2,1,1,1,1,1,1,1,1,1,1] => 121
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
-----------------------------------------------------------------------------
Created: Sep 15, 2015 at 08:40 by Martin Rubey
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Last Updated: Jul 12, 2017 at 10:03 by Christian Stump