- FindStat

Identifier

St000284: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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
                    

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$F_{2} = 2\ q$

$F_{3} = 2\ q + q^{4}$

$F_{4} = 2\ q + q^{4} + 2\ q^{9}$

$F_{5} = 2\ q + 2\ q^{16} + 2\ q^{25} + q^{36}$

$F_{6} = 2\ q + 4\ q^{25} + 2\ q^{81} + 2\ q^{100} + q^{256}$

$F_{7} = 2\ q + 2\ q^{36} + 4\ q^{196} + 2\ q^{225} + q^{400} + 2\ q^{441} + 2\ q^{1225}$

$F_{8} = 2\ q + 2\ q^{49} + 2\ q^{196} + 2\ q^{400} + 2\ q^{441} + 2\ q^{784} + 2\ q^{1225} + q^{1764} + 2\ q^{3136} + 2\ q^{4096} + 2\ q^{4900} + q^{8100}$

$F_{9} = 2\ q + 2\ q^{64} + 2\ q^{729} + 2\ q^{784} + 3\ q^{1764} + 2\ q^{2304} + 2\ q^{3136} + q^{4900} + 2\ q^{7056} + 2\ q^{11025} + 2\ q^{14400} + 2\ q^{26244} + 2\ q^{28224} + 2\ q^{35721} + 2\ q^{46656}$

$F_{10} = 2\ q + 2\ q^{81} + 2\ q^{1225} + 2\ q^{1296} + 2\ q^{1764} + 2\ q^{5625} + 2\ q^{7056} + 2\ q^{8100} + 2\ q^{15876} + 2\ q^{25600} + 2\ q^{44100} + 2\ q^{50625} + 2\ q^{63504} + 2\ q^{82944} + 2\ q^{90000} + 2\ q^{99225} + 2\ q^{122500} + q^{200704} + 2\ q^{202500} + 2\ q^{275625} + 2\ q^{321489} + q^{589824}$

$F_{11} = 2\ q + 2\ q^{100} + 2\ q^{1936} + 2\ q^{2025} + 2\ q^{12100} + 2\ q^{14400} + 2\ q^{17424} + 2\ q^{27225} + 2\ q^{44100} + 2\ q^{53361} + q^{63504} + 2\ q^{108900} + 2\ q^{148225} + 2\ q^{213444} + 2\ q^{302500} + 2\ q^{352836} + 2\ q^{435600} + 2\ q^{480249} + 2\ q^{680625} + 2\ q^{853776} + 4\ q^{980100} + 2\ q^{1210000} + 2\ q^{1334025} + q^{1411344} + 2\ q^{1517824} + 2\ q^{1742400} + 2\ q^{2371600} + 2\ q^{5336100}$

$F_{12} = 2\ q + 2\ q^{121} + 2\ q^{2916} + 2\ q^{3025} + 2\ q^{17424} + 2\ q^{23716} + 2\ q^{27225} + 2\ q^{75625} + 2\ q^{88209} + 2\ q^{102400} + 2\ q^{108900} + 4\ q^{213444} + 2\ q^{379456} + 2\ q^{793881} + 2\ q^{893025} + 2\ q^{1334025} + 2\ q^{1742400} + 2\ q^{1982464} + 2\ q^{2205225} + 2\ q^{2722500} + 2\ q^{2985984} + 4\ q^{3705625} + 2\ q^{4322241} + q^{4410000} + 2\ q^{4460544} + 2\ q^{5645376} + q^{6969600} + 2\ q^{7144929} + 2\ q^{8820900} + 2\ q^{9486400} + 2\ q^{12390400} + 2\ q^{12702096} + 2\ q^{13660416} + 2\ q^{17288964} + 2\ q^{19847025} + 2\ q^{31719424} + 2\ q^{33350625} + q^{59290000}$

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

Created

Sep 15, 2015 at 08:40 by Martin Rubey

Updated

Jul 12, 2017 at 10:03 by Christian Stump