- FindStat

Identifier

St000818: Integer compositions ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1,1]=>1
[2]=>2
[1,1,1]=>1
[1,2]=>2
[2,1]=>2
[3]=>4
[1,1,1,1]=>1
[1,1,2]=>2
[1,2,1]=>2
[1,3]=>8
[2,1,1]=>2
[2,2]=>6
[3,1]=>4
[4]=>8
[1,1,1,1,1]=>1
[1,1,1,2]=>2
[1,1,2,1]=>2
[1,1,3]=>12
[1,2,1,1]=>2
[1,2,2]=>6
[1,3,1]=>8
[1,4]=>24
[2,1,1,1]=>2
[2,1,2]=>4
[2,2,1]=>6
[2,3]=>12
[3,1,1]=>4
[3,2]=>16
[4,1]=>8
[5]=>16
[1,1,1,1,1,1]=>1
[1,1,1,1,2]=>2
[1,1,1,2,1]=>2
[1,1,1,3]=>16
[1,1,2,1,1]=>2
[1,1,2,2]=>6
[1,1,3,1]=>12
[1,1,4]=>48
[1,2,1,1,1]=>2
[1,2,1,2]=>4
[1,2,2,1]=>6
[1,2,3]=>12
[1,3,1,1]=>8
[1,3,2]=>28
[1,4,1]=>24
[1,5]=>64
[2,1,1,1,1]=>2
[2,1,1,2]=>4
[2,1,2,1]=>4
[2,1,3]=>20
[2,2,1,1]=>6
[2,2,2]=>22
[2,3,1]=>12
[2,4]=>56
[3,1,1,1]=>4
[3,1,2]=>8
[3,2,1]=>16
[3,3]=>44
[4,1,1]=>8
[4,2]=>40
[5,1]=>16
[6]=>32
[1,1,1,1,1,1,1]=>1
[1,1,1,1,1,2]=>2
[1,1,1,1,2,1]=>2
[1,1,1,1,3]=>20
[1,1,1,2,1,1]=>2
[1,1,1,2,2]=>6
[1,1,1,3,1]=>16
[1,1,1,4]=>80
[1,1,2,1,1,1]=>2
[1,1,2,1,2]=>4
[1,1,2,2,1]=>6
[1,1,2,3]=>12
[1,1,3,1,1]=>12
[1,1,3,2]=>40
[1,1,4,1]=>48
[1,1,5]=>160
[1,2,1,1,1,1]=>2
[1,2,1,1,2]=>4
[1,2,1,2,1]=>4
[1,2,1,3]=>20
[1,2,2,1,1]=>6
[1,2,2,2]=>22
[1,2,3,1]=>12
[1,2,4]=>80
[1,3,1,1,1]=>8
[1,3,1,2]=>16
[1,3,2,1]=>28
[1,3,3]=>112
[1,4,1,1]=>24
[1,4,2]=>96
[1,5,1]=>64
[1,6]=>160
[2,1,1,1,1,1]=>2
[2,1,1,1,2]=>4
[2,1,1,2,1]=>4
[2,1,1,3]=>28
[2,1,2,1,1]=>4
[2,1,2,2]=>12
[2,1,3,1]=>20
[2,1,4]=>112
[2,2,1,1,1]=>6
[2,2,1,2]=>12
[2,2,2,1]=>22
[2,2,3]=>44
[2,3,1,1]=>12
[2,3,2]=>56
[2,4,1]=>56
[2,5]=>192
[3,1,1,1,1]=>4
[3,1,1,2]=>8
[3,1,2,1]=>8
[3,1,3]=>48
[3,2,1,1]=>16
[3,2,2]=>68
[3,3,1]=>44
[3,4]=>88
[4,1,1,1]=>8
[4,1,2]=>16
[4,2,1]=>40
[4,3]=>136
[5,1,1]=>16
[5,2]=>96
[6,1]=>32
[7]=>64
[1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,2]=>2
[1,1,1,1,1,2,1]=>2
[1,1,1,1,1,3]=>24
[1,1,1,1,2,1,1]=>2
[1,1,1,1,2,2]=>6
[1,1,1,1,3,1]=>20
[1,1,1,1,4]=>120
[1,1,1,2,1,1,1]=>2
[1,1,1,2,1,2]=>4
[1,1,1,2,2,1]=>6
[1,1,1,2,3]=>12
[1,1,1,3,1,1]=>16
[1,1,1,3,2]=>52
[1,1,1,4,1]=>80
[1,1,1,5]=>320
[1,1,2,1,1,1,1]=>2
[1,1,2,1,1,2]=>4
[1,1,2,1,2,1]=>4
[1,1,2,1,3]=>20
[1,1,2,2,1,1]=>6
[1,1,2,2,2]=>22
[1,1,2,3,1]=>12
[1,1,2,4]=>104
[1,1,3,1,1,1]=>12
[1,1,3,1,2]=>24
[1,1,3,2,1]=>40
[1,1,3,3]=>204
[1,1,4,1,1]=>48
[1,1,4,2]=>176
[1,1,5,1]=>160
[1,1,6]=>480
[1,2,1,1,1,1,1]=>2
[1,2,1,1,1,2]=>4
[1,2,1,1,2,1]=>4
[1,2,1,1,3]=>28
[1,2,1,2,1,1]=>4
[1,2,1,2,2]=>12
[1,2,1,3,1]=>20
[1,2,1,4]=>152
[1,2,2,1,1,1]=>6
[1,2,2,1,2]=>12
[1,2,2,2,1]=>22
[1,2,2,3]=>44
[1,2,3,1,1]=>12
[1,2,3,2]=>56
[1,2,4,1]=>80
[1,2,5]=>352
[1,3,1,1,1,1]=>8
[1,3,1,1,2]=>16
[1,3,1,2,1]=>16
[1,3,1,3]=>112
[1,3,2,1,1]=>28
[1,3,2,2]=>112
[1,3,3,1]=>112
[1,3,4]=>224
[1,4,1,1,1]=>24
[1,4,1,2]=>48
[1,4,2,1]=>96
[1,4,3]=>496
[1,5,1,1]=>64
[1,5,2]=>288
[1,6,1]=>160
[1,7]=>384
[2,1,1,1,1,1,1]=>2
[2,1,1,1,1,2]=>4
[2,1,1,1,2,1]=>4
[2,1,1,1,3]=>36
[2,1,1,2,1,1]=>4
[2,1,1,2,2]=>12
[2,1,1,3,1]=>28
[2,1,1,4]=>184
[2,1,2,1,1,1]=>4
[2,1,2,1,2]=>8
[2,1,2,2,1]=>12
[2,1,2,3]=>24
[2,1,3,1,1]=>20
[2,1,3,2]=>68
[2,1,4,1]=>112
[2,1,5]=>448
[2,2,1,1,1,1]=>6
[2,2,1,1,2]=>12
[2,2,1,2,1]=>12
[2,2,1,3]=>68
[2,2,2,1,1]=>22
[2,2,2,2]=>90
[2,2,3,1]=>44
[2,2,4]=>336
[2,3,1,1,1]=>12
[2,3,1,2]=>24
[2,3,2,1]=>56
[2,3,3]=>156
[2,4,1,1]=>56
[2,4,2]=>304
[2,5,1]=>192
[2,6]=>576
[3,1,1,1,1,1]=>4
[3,1,1,1,2]=>8
[3,1,1,2,1]=>8
[3,1,1,3]=>64
[3,1,2,1,1]=>8
[3,1,2,2]=>24
[3,1,3,1]=>48
[3,1,4]=>184
[3,2,1,1,1]=>16
[3,2,1,2]=>32
[3,2,2,1]=>68
[3,2,3]=>136
[3,3,1,1]=>44
[3,3,2]=>248
[3,4,1]=>88
[3,5]=>448
[4,1,1,1,1]=>8
[4,1,1,2]=>16
[4,1,2,1]=>16
[4,1,3]=>112
[4,2,1,1]=>40
[4,2,2]=>192
[4,3,1]=>136
[4,4]=>360
[5,1,1,1]=>16
[5,1,2]=>32
[5,2,1]=>96
[5,3]=>384
[6,1,1]=>32
[6,2]=>224
[7,1]=>64
[8]=>128
[1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,2]=>2
[1,1,1,1,1,1,2,1]=>2
[1,1,1,1,1,1,3]=>28
[1,1,1,1,1,2,1,1]=>2
[1,1,1,1,1,2,2]=>6
[1,1,1,1,1,3,1]=>24
[1,1,1,1,1,4]=>168
[1,1,1,1,2,1,1,1]=>2
[1,1,1,1,2,1,2]=>4
[1,1,1,1,2,2,1]=>6
[1,1,1,1,2,3]=>12
[1,1,1,1,3,1,1]=>20
[1,1,1,1,3,2]=>64
[1,1,1,1,4,1]=>120
[1,1,1,1,5]=>560
[1,1,1,2,1,1,1,1]=>2
[1,1,1,2,1,1,2]=>4
[1,1,1,2,1,2,1]=>4
[1,1,1,2,1,3]=>20
[1,1,1,2,2,1,1]=>6
[1,1,1,2,2,2]=>22
[1,1,1,2,3,1]=>12
[1,1,1,2,4]=>128
[1,1,1,3,1,1,1]=>16
[1,1,1,3,1,2]=>32
[1,1,1,3,2,1]=>52
[1,1,1,3,3]=>320
[1,1,1,4,1,1]=>80
[1,1,1,4,2]=>280
[1,1,1,5,1]=>320
[1,1,1,6]=>1120
[1,1,2,1,1,1,1,1]=>2
[1,1,2,1,1,1,2]=>4
[1,1,2,1,1,2,1]=>4
[1,1,2,1,1,3]=>28
[1,1,2,1,2,1,1]=>4
[1,1,2,1,2,2]=>12
[1,1,2,1,3,1]=>20
[1,1,2,1,4]=>192
[1,1,2,2,1,1,1]=>6
[1,1,2,2,1,2]=>12
[1,1,2,2,2,1]=>22
[1,1,2,2,3]=>44
[1,1,2,3,1,1]=>12
[1,1,2,3,2]=>56
[1,1,2,4,1]=>104
[1,1,2,5]=>560
[1,1,3,1,1,1,1]=>12
[1,1,3,1,1,2]=>24
[1,1,3,1,2,1]=>24
[1,1,3,1,3]=>192
[1,1,3,2,1,1]=>40
[1,1,3,2,2]=>156
[1,1,3,3,1]=>204
[1,1,3,4]=>408
[1,1,4,1,1,1]=>48
[1,1,4,1,2]=>96
[1,1,4,2,1]=>176
[1,1,4,3]=>1176
[1,1,5,1,1]=>160
[1,1,5,2]=>640
[1,1,6,1]=>480
[1,1,7]=>1344
[1,2,1,1,1,1,1,1]=>2
[1,2,1,1,1,1,2]=>4
[1,2,1,1,1,2,1]=>4
[1,2,1,1,1,3]=>36
[1,2,1,1,2,1,1]=>4
[1,2,1,1,2,2]=>12
[1,2,1,1,3,1]=>28
[1,2,1,1,4]=>240
[1,2,1,2,1,1,1]=>4
[1,2,1,2,1,2]=>8
[1,2,1,2,2,1]=>12
[1,2,1,2,3]=>24
[1,2,1,3,1,1]=>20
[1,2,1,3,2]=>68
[1,2,1,4,1]=>152
[1,2,1,5]=>752
[1,2,2,1,1,1,1]=>6
[1,2,2,1,1,2]=>12
[1,2,2,1,2,1]=>12
[1,2,2,1,3]=>68
[1,2,2,2,1,1]=>22
[1,2,2,2,2]=>90
[1,2,2,3,1]=>44
[1,2,2,4]=>424
[1,2,3,1,1,1]=>12
[1,2,3,1,2]=>24
[1,2,3,2,1]=>56
[1,2,3,3]=>156
[1,2,4,1,1]=>80
[1,2,4,2]=>416
[1,2,5,1]=>352
[1,2,6]=>1280
[1,3,1,1,1,1,1]=>8
[1,3,1,1,1,2]=>16
[1,3,1,1,2,1]=>16
[1,3,1,1,3]=>144
[1,3,1,2,1,1]=>16
[1,3,1,2,2]=>48
[1,3,1,3,1]=>112
[1,3,1,4]=>448
[1,3,2,1,1,1]=>28
[1,3,2,1,2]=>56
[1,3,2,2,1]=>112
[1,3,2,3]=>224
[1,3,3,1,1]=>112
[1,3,3,2]=>540
[1,3,4,1]=>224
[1,3,5]=>1440
[1,4,1,1,1,1]=>24
[1,4,1,1,2]=>48
[1,4,1,2,1]=>48
[1,4,1,3]=>400
[1,4,2,1,1]=>96
[1,4,2,2]=>416
[1,4,3,1]=>496
[1,4,4]=>1576
[1,5,1,1,1]=>64
[1,5,1,2]=>128
[1,5,2,1]=>288
[1,5,3]=>1760
[1,6,1,1]=>160
[1,6,2]=>800
[1,7,1]=>384
[1,8]=>896
[2,1,1,1,1,1,1,1]=>2
[2,1,1,1,1,1,2]=>4
[2,1,1,1,1,2,1]=>4
[2,1,1,1,1,3]=>44
[2,1,1,1,2,1,1]=>4
[2,1,1,1,2,2]=>12
[2,1,1,1,3,1]=>36
[2,1,1,1,4]=>272
[2,1,1,2,1,1,1]=>4
[2,1,1,2,1,2]=>8
[2,1,1,2,2,1]=>12
[2,1,1,2,3]=>24
[2,1,1,3,1,1]=>28
[2,1,1,3,2]=>92
[2,1,1,4,1]=>184
[2,1,1,5]=>848
[2,1,2,1,1,1,1]=>4
[2,1,2,1,1,2]=>8
[2,1,2,1,2,1]=>8
[2,1,2,1,3]=>40
[2,1,2,2,1,1]=>12
[2,1,2,2,2]=>44
[2,1,2,3,1]=>24
[2,1,2,4]=>232
[2,1,3,1,1,1]=>20
[2,1,3,1,2]=>40
[2,1,3,2,1]=>68
[2,1,3,3]=>316
[2,1,4,1,1]=>112
[2,1,4,2]=>408
[2,1,5,1]=>448
[2,1,6]=>1536
[2,2,1,1,1,1,1]=>6
[2,2,1,1,1,2]=>12
[2,2,1,1,2,1]=>12
[2,2,1,1,3]=>92
[2,2,1,2,1,1]=>12
[2,2,1,2,2]=>36
[2,2,1,3,1]=>68
[2,2,1,4]=>584
[2,2,2,1,1,1]=>22
[2,2,2,1,2]=>44
[2,2,2,2,1]=>90
[2,2,2,3]=>180
[2,2,3,1,1]=>44
[2,2,3,2]=>224
[2,2,4,1]=>336
[2,2,5]=>1664
[2,3,1,1,1,1]=>12
[2,3,1,1,2]=>24
[2,3,1,2,1]=>24
[2,3,1,3]=>160
[2,3,2,1,1]=>56
[2,3,2,2]=>260
[2,3,3,1]=>156
[2,3,4]=>312
[2,4,1,1,1]=>56
[2,4,1,2]=>112
[2,4,2,1]=>304
[2,4,3]=>944
[2,5,1,1]=>192
[2,5,2]=>1184
[2,6,1]=>576
[2,7]=>1600
[3,1,1,1,1,1,1]=>4
[3,1,1,1,1,2]=>8
[3,1,1,1,2,1]=>8
[3,1,1,1,3]=>80
[3,1,1,2,1,1]=>8
[3,1,1,2,2]=>24
[3,1,1,3,1]=>64
[3,1,1,4]=>312
[3,1,2,1,1,1]=>8
[3,1,2,1,2]=>16
[3,1,2,2,1]=>24
[3,1,2,3]=>48
[3,1,3,1,1]=>48
[3,1,3,2]=>160
[3,1,4,1]=>184
[3,1,5]=>1040
[3,2,1,1,1,1]=>16
[3,2,1,1,2]=>32
[3,2,1,2,1]=>32
[3,2,1,3]=>200
[3,2,2,1,1]=>68
[3,2,2,2]=>304
[3,2,3,1]=>136
[3,2,4]=>768
[3,3,1,1,1]=>44
[3,3,1,2]=>88
[3,3,2,1]=>248
[3,3,3]=>788
[3,4,1,1]=>88
[3,4,2]=>584
[3,5,1]=>448
[3,6]=>1664
[4,1,1,1,1,1]=>8
[4,1,1,1,2]=>16
[4,1,1,2,1]=>16
[4,1,1,3]=>144
[4,1,2,1,1]=>16
[4,1,2,2]=>48
[4,1,3,1]=>112
[4,1,4]=>496
[4,2,1,1,1]=>40
[4,2,1,2]=>80
[4,2,2,1]=>192
[4,2,3]=>384
[4,3,1,1]=>136
[4,3,2]=>880
[4,4,1]=>360
[4,5]=>720
[5,1,1,1,1]=>16
[5,1,1,2]=>32
[5,1,2,1]=>32
[5,1,3]=>256
[5,2,1,1]=>96
[5,2,2]=>512
[5,3,1]=>384
[5,4]=>1216
[6,1,1,1]=>32
[6,1,2]=>64
[6,2,1]=>224
[6,3]=>1024
[7,1,1]=>64
[7,2]=>512
[8,1]=>128
[9]=>256
                    

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$F_{2} = q + q^{2}$

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

$F_{4} = q + 3\ q^{2} + q^{4} + q^{6} + 2\ q^{8}$

$F_{5} = q + 4\ q^{2} + 2\ q^{4} + 2\ q^{6} + 2\ q^{8} + 2\ q^{12} + 2\ q^{16} + q^{24}$

$F_{6} = q + 5\ q^{2} + 4\ q^{4} + 3\ q^{6} + 3\ q^{8} + 3\ q^{12} + 3\ q^{16} + q^{20} + q^{22} + q^{24} + q^{28} + q^{32} + q^{40} + q^{44} + q^{48} + q^{56} + q^{64}$

$F_{7} = q + 6\ q^{2} + 7\ q^{4} + 4\ q^{6} + 4\ q^{8} + 6\ q^{12} + 5\ q^{16} + 3\ q^{20} + 2\ q^{22} + q^{24} + 2\ q^{28} + q^{32} + 2\ q^{40} + 2\ q^{44} + 2\ q^{48} + 2\ q^{56} + 2\ q^{64} + q^{68} + 2\ q^{80} + q^{88} + 2\ q^{96} + 2\ q^{112} + q^{136} + 2\ q^{160} + q^{192}$

$F_{8} = q + 7\ q^{2} + 11\ q^{4} + 5\ q^{6} + 6\ q^{8} + 11\ q^{12} + 7\ q^{16} + 4\ q^{20} + 3\ q^{22} + 6\ q^{24} + 3\ q^{28} + 3\ q^{32} + q^{36} + 2\ q^{40} + 3\ q^{44} + 3\ q^{48} + q^{52} + 3\ q^{56} + 3\ q^{64} + 3\ q^{68} + 2\ q^{80} + q^{88} + q^{90} + 2\ q^{96} + q^{104} + 5\ q^{112} + q^{120} + q^{128} + 2\ q^{136} + q^{152} + q^{156} + 2\ q^{160} + q^{176} + 2\ q^{184} + 2\ q^{192} + q^{204} + 2\ q^{224} + q^{248} + q^{288} + q^{304} + q^{320} + q^{336} + q^{352} + q^{360} + 2\ q^{384} + 2\ q^{448} + q^{480} + q^{496} + q^{576}$

$F_{9} = q + 8\ q^{2} + 16\ q^{4} + 6\ q^{6} + 10\ q^{8} + 18\ q^{12} + 10\ q^{16} + 5\ q^{20} + 4\ q^{22} + 12\ q^{24} + 5\ q^{28} + 6\ q^{32} + 3\ q^{36} + 4\ q^{40} + 7\ q^{44} + 7\ q^{48} + q^{52} + 5\ q^{56} + 5\ q^{64} + 5\ q^{68} + 4\ q^{80} + 2\ q^{88} + 2\ q^{90} + 2\ q^{92} + 3\ q^{96} + q^{104} + 6\ q^{112} + q^{120} + 3\ q^{128} + 2\ q^{136} + 2\ q^{144} + q^{152} + 3\ q^{156} + 4\ q^{160} + q^{168} + q^{176} + q^{180} + 2\ q^{184} + 4\ q^{192} + q^{200} + q^{204} + 4\ q^{224} + q^{232} + q^{240} + q^{248} + 2\ q^{256} + q^{260} + q^{272} + q^{280} + q^{288} + 2\ q^{304} + 2\ q^{312} + q^{316} + 2\ q^{320} + q^{336} + q^{352} + q^{360} + 3\ q^{384} + q^{400} + 2\ q^{408} + 2\ q^{416} + q^{424} + 3\ q^{448} + q^{480} + 2\ q^{496} + 2\ q^{512} + q^{540} + 2\ q^{560} + q^{576} + 2\ q^{584} + q^{640} + q^{720} + q^{752} + q^{768} + q^{788} + q^{800} + q^{848} + q^{880} + q^{896} + q^{944} + q^{1024} + q^{1040} + q^{1120} + q^{1176} + q^{1184} + q^{1216} + q^{1280} + q^{1344} + q^{1440} + q^{1536} + q^{1576} + q^{1600} + 2\ q^{1664} + q^{1760}$

Description

The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions.
For example, $QS_{31} = M_{1111} + M_{121} + M_{211} + M_{31}$, so the statistic on the composition $31$ is 4.
Apparently, the sum over all compositions gives the sequence oeis:A138178.

Code

def statistic(mu):
    M = QuasiSymmetricFunctions(ZZ).M()
    QS = QuasiSymmetricFunctions(ZZ).QS()
    return sum(coeff for _, coeff in M(QS(mu)))

Created

May 20, 2017 at 21:59 by Martin Rubey

Updated

May 20, 2017 at 22:26 by Martin Rubey