- FindStat

Identifier

St001968: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>1
[2]=>1
[1,1]=>1
[3]=>1
[2,1]=>3
[1,1,1]=>2
[4]=>1
[3,1]=>4
[2,2]=>3
[2,1,1]=>12
[1,1,1,1]=>6
[5]=>1
[4,1]=>5
[3,2]=>10
[3,1,1]=>20
[2,2,1]=>30
[2,1,1,1]=>60
[1,1,1,1,1]=>24
[6]=>1
[5,1]=>6
[4,2]=>15
[4,1,1]=>30
[3,3]=>10
[3,2,1]=>120
[3,1,1,1]=>120
[2,2,2]=>30
[2,2,1,1]=>270
[2,1,1,1,1]=>360
[1,1,1,1,1,1]=>120
[7]=>1
[6,1]=>7
[5,2]=>21
[5,1,1]=>42
[4,3]=>35
[4,2,1]=>210
[4,1,1,1]=>210
[3,3,1]=>140
[3,2,2]=>210
[3,2,1,1]=>1260
[3,1,1,1,1]=>840
[2,2,2,1]=>630
[2,2,1,1,1]=>2520
[2,1,1,1,1,1]=>2520
[1,1,1,1,1,1,1]=>720
[8]=>1
[7,1]=>8
[6,2]=>28
[6,1,1]=>56
[5,3]=>56
[5,2,1]=>336
[5,1,1,1]=>336
[4,4]=>35
[4,3,1]=>560
[4,2,2]=>420
[4,2,1,1]=>2520
[4,1,1,1,1]=>1680
[3,3,2]=>560
[3,3,1,1]=>1680
[3,2,2,1]=>5040
[3,2,1,1,1]=>13440
[3,1,1,1,1,1]=>6720
[2,2,2,2]=>630
[2,2,2,1,1]=>10080
[2,2,1,1,1,1]=>25200
[2,1,1,1,1,1,1]=>20160
[1,1,1,1,1,1,1,1]=>5040
[9]=>1
[8,1]=>9
[7,2]=>36
[7,1,1]=>72
[6,3]=>84
[6,2,1]=>504
[6,1,1,1]=>504
[5,4]=>126
[5,3,1]=>1008
[5,2,2]=>756
[5,2,1,1]=>4536
[5,1,1,1,1]=>3024
[4,4,1]=>630
[4,3,2]=>2520
[4,3,1,1]=>7560
[4,2,2,1]=>11340
[4,2,1,1,1]=>30240
[4,1,1,1,1,1]=>15120
[3,3,3]=>560
[3,3,2,1]=>15120
[3,3,1,1,1]=>20160
[3,2,2,2]=>7560
[3,2,2,1,1]=>90720
[3,2,1,1,1,1]=>151200
[3,1,1,1,1,1,1]=>60480
[2,2,2,2,1]=>22680
[2,2,2,1,1,1]=>151200
[2,2,1,1,1,1,1]=>272160
[2,1,1,1,1,1,1,1]=>181440
[1,1,1,1,1,1,1,1,1]=>40320
[10]=>1
[9,1]=>10
[8,2]=>45
[8,1,1]=>90
[7,3]=>120
[7,2,1]=>720
[7,1,1,1]=>720
[6,4]=>210
[6,3,1]=>1680
[6,2,2]=>1260
[6,2,1,1]=>7560
[6,1,1,1,1]=>5040
[5,5]=>126
[5,4,1]=>2520
[5,3,2]=>5040
[5,3,1,1]=>15120
[5,2,2,1]=>22680
[5,2,1,1,1]=>60480
[5,1,1,1,1,1]=>30240
[4,4,2]=>3150
[4,4,1,1]=>9450
[4,3,3]=>4200
[4,3,2,1]=>75600
[4,3,1,1,1]=>100800
[4,2,2,2]=>18900
[4,2,2,1,1]=>226800
[4,2,1,1,1,1]=>378000
[4,1,1,1,1,1,1]=>151200
[3,3,3,1]=>16800
[3,3,2,2]=>37800
[3,3,2,1,1]=>302400
[3,3,1,1,1,1]=>252000
[3,2,2,2,1]=>302400
[3,2,2,1,1,1]=>1512000
[3,2,1,1,1,1,1]=>1814400
[3,1,1,1,1,1,1,1]=>604800
[2,2,2,2,2]=>22680
[2,2,2,2,1,1]=>567000
[2,2,2,1,1,1,1]=>2268000
[2,2,1,1,1,1,1,1]=>3175200
[2,1,1,1,1,1,1,1,1]=>1814400
[1,1,1,1,1,1,1,1,1,1]=>362880
[11]=>1
[10,1]=>11
[9,2]=>55
[9,1,1]=>110
[8,3]=>165
[8,2,1]=>990
[8,1,1,1]=>990
[7,4]=>330
[7,3,1]=>2640
[7,2,2]=>1980
[7,2,1,1]=>11880
[7,1,1,1,1]=>7920
[6,5]=>462
[6,4,1]=>4620
[6,3,2]=>9240
[6,3,1,1]=>27720
[6,2,2,1]=>41580
[6,2,1,1,1]=>110880
[6,1,1,1,1,1]=>55440
[5,5,1]=>2772
[5,4,2]=>13860
[5,4,1,1]=>41580
[5,3,3]=>9240
[5,3,2,1]=>166320
[5,3,1,1,1]=>221760
[5,2,2,2]=>41580
[5,2,2,1,1]=>498960
[5,2,1,1,1,1]=>831600
[5,1,1,1,1,1,1]=>332640
[4,4,3]=>11550
[4,4,2,1]=>103950
[4,4,1,1,1]=>138600
[4,3,3,1]=>138600
[4,3,2,2]=>207900
[4,3,2,1,1]=>1663200
[4,3,1,1,1,1]=>1386000
[4,2,2,2,1]=>831600
[4,2,2,1,1,1]=>4158000
[4,2,1,1,1,1,1]=>4989600
[4,1,1,1,1,1,1,1]=>1663200
[3,3,3,2]=>92400
[3,3,3,1,1]=>369600
[3,3,2,2,1]=>1663200
[3,3,2,1,1,1]=>5544000
[3,3,1,1,1,1,1]=>3326400
[3,2,2,2,2]=>415800
[3,2,2,2,1,1]=>8316000
[3,2,2,1,1,1,1]=>24948000
[3,2,1,1,1,1,1,1]=>23284800
[3,1,1,1,1,1,1,1,1]=>6652800
[2,2,2,2,2,1]=>1247400
[2,2,2,2,1,1,1]=>12474000
[2,2,2,1,1,1,1,1]=>34927200
[2,2,1,1,1,1,1,1,1]=>39916800
[2,1,1,1,1,1,1,1,1,1]=>19958400
[1,1,1,1,1,1,1,1,1,1,1]=>3628800
[12]=>1
[11,1]=>12
[10,2]=>66
[10,1,1]=>132
[9,3]=>220
[9,2,1]=>1320
[9,1,1,1]=>1320
[8,4]=>495
[8,3,1]=>3960
[8,2,2]=>2970
[8,2,1,1]=>17820
[8,1,1,1,1]=>11880
[7,5]=>792
[7,4,1]=>7920
[7,3,2]=>15840
[7,3,1,1]=>47520
[7,2,2,1]=>71280
[7,2,1,1,1]=>190080
[7,1,1,1,1,1]=>95040
[6,6]=>462
[6,5,1]=>11088
[6,4,2]=>27720
[6,4,1,1]=>83160
[6,3,3]=>18480
[6,3,2,1]=>332640
[6,3,1,1,1]=>443520
[6,2,2,2]=>83160
[6,2,2,1,1]=>997920
[6,2,1,1,1,1]=>1663200
[6,1,1,1,1,1,1]=>665280
[5,5,2]=>16632
[5,5,1,1]=>49896
[5,4,3]=>55440
[5,4,2,1]=>498960
[5,4,1,1,1]=>665280
[5,3,3,1]=>332640
[5,3,2,2]=>498960
[5,3,2,1,1]=>3991680
[5,3,1,1,1,1]=>3326400
[5,2,2,2,1]=>1995840
[5,2,2,1,1,1]=>9979200
[5,2,1,1,1,1,1]=>11975040
[5,1,1,1,1,1,1,1]=>3991680
[4,4,4]=>11550
[4,4,3,1]=>415800
[4,4,2,2]=>311850
[4,4,2,1,1]=>2494800
[4,4,1,1,1,1]=>2079000
[4,3,3,2]=>831600
[4,3,3,1,1]=>3326400
[4,3,2,2,1]=>9979200
[4,3,2,1,1,1]=>33264000
[4,3,1,1,1,1,1]=>19958400
[4,2,2,2,2]=>1247400
[4,2,2,2,1,1]=>24948000
[4,2,2,1,1,1,1]=>74844000
[4,2,1,1,1,1,1,1]=>69854400
[4,1,1,1,1,1,1,1,1]=>19958400
[3,3,3,3]=>92400
[3,3,3,2,1]=>4435200
[3,3,3,1,1,1]=>7392000
[3,3,2,2,2]=>3326400
[3,3,2,2,1,1]=>49896000
[3,3,2,1,1,1,1]=>99792000
[3,3,1,1,1,1,1,1]=>46569600
[3,2,2,2,2,1]=>24948000
[3,2,2,2,1,1,1]=>199584000
[3,2,2,1,1,1,1,1]=>419126400
[3,2,1,1,1,1,1,1,1]=>319334400
[3,1,1,1,1,1,1,1,1,1]=>79833600
[2,2,2,2,2,2]=>1247400
[2,2,2,2,2,1,1]=>44906400
[2,2,2,2,1,1,1,1]=>261954000
[2,2,2,1,1,1,1,1,1]=>558835200
[2,2,1,1,1,1,1,1,1,1]=>538876800
[2,1,1,1,1,1,1,1,1,1,1]=>239500800
[1,1,1,1,1,1,1,1,1,1,1,1]=>39916800
[13]=>1
[12,1]=>13
[11,2]=>78
[11,1,1]=>156
[10,3]=>286
[10,2,1]=>1716
[10,1,1,1]=>1716
[9,4]=>715
[9,3,1]=>5720
[9,2,2]=>4290
[9,2,1,1]=>25740
[9,1,1,1,1]=>17160
[8,5]=>1287
[8,4,1]=>12870
[8,3,2]=>25740
[8,3,1,1]=>77220
[8,2,2,1]=>115830
[8,2,1,1,1]=>308880
[8,1,1,1,1,1]=>154440
[7,6]=>1716
[7,5,1]=>20592
[7,4,2]=>51480
[7,4,1,1]=>154440
[7,3,3]=>34320
[7,3,2,1]=>617760
[7,3,1,1,1]=>823680
[7,2,2,2]=>154440
[7,2,2,1,1]=>1853280
[7,2,1,1,1,1]=>3088800
[7,1,1,1,1,1,1]=>1235520
[6,6,1]=>12012
[6,5,2]=>72072
[6,5,1,1]=>216216
[6,4,3]=>120120
[6,4,2,1]=>1081080
[6,4,1,1,1]=>1441440
[6,3,3,1]=>720720
[6,3,2,2]=>1081080
[6,3,2,1,1]=>8648640
[6,3,1,1,1,1]=>7207200
[6,2,2,2,1]=>4324320
[6,2,2,1,1,1]=>21621600
[6,2,1,1,1,1,1]=>25945920
[6,1,1,1,1,1,1,1]=>8648640
[5,5,3]=>72072
[5,5,2,1]=>648648
[5,5,1,1,1]=>864864
[5,4,4]=>90090
[5,4,3,1]=>2162160
[5,4,2,2]=>1621620
[5,4,2,1,1]=>12972960
[5,4,1,1,1,1]=>10810800
[5,3,3,2]=>2162160
[5,3,3,1,1]=>8648640
[5,3,2,2,1]=>25945920
[5,3,2,1,1,1]=>86486400
[5,3,1,1,1,1,1]=>51891840
[5,2,2,2,2]=>3243240
[5,2,2,2,1,1]=>64864800
[5,2,2,1,1,1,1]=>194594400
[5,2,1,1,1,1,1,1]=>181621440
[5,1,1,1,1,1,1,1,1]=>51891840
[4,4,4,1]=>450450
[4,4,3,2]=>2702700
[4,4,3,1,1]=>10810800
[4,4,2,2,1]=>16216200
[4,4,2,1,1,1]=>54054000
[4,4,1,1,1,1,1]=>32432400
[4,3,3,3]=>1201200
[4,3,3,2,1]=>43243200
[4,3,3,1,1,1]=>72072000
[4,3,2,2,2]=>21621600
[4,3,2,2,1,1]=>324324000
[4,3,2,1,1,1,1]=>648648000
[4,3,1,1,1,1,1,1]=>302702400
[4,2,2,2,2,1]=>81081000
[4,2,2,2,1,1,1]=>648648000
[4,2,2,1,1,1,1,1]=>1362160800
[4,2,1,1,1,1,1,1,1]=>1037836800
[4,1,1,1,1,1,1,1,1,1]=>259459200
[3,3,3,3,1]=>4804800
[3,3,3,2,2]=>14414400
[3,3,3,2,1,1]=>144144000
[3,3,3,1,1,1,1]=>144144000
[3,3,2,2,2,1]=>216216000
[3,3,2,2,1,1,1]=>1297296000
[3,3,2,1,1,1,1,1]=>1816214400
[3,3,1,1,1,1,1,1,1]=>691891200
[3,2,2,2,2,2]=>32432400
[3,2,2,2,2,1,1]=>972972000
[3,1,1,1,1,1,1,1,1,1,1]=>1037836800
[2,2,2,2,2,2,1]=>97297200
[2,2,2,2,2,1,1,1]=>1362160800
[1,1,1,1,1,1,1,1,1,1,1,1,1]=>479001600
[14]=>1
[13,1]=>14
[12,2]=>91
[12,1,1]=>182
[11,3]=>364
[11,2,1]=>2184
[11,1,1,1]=>2184
[10,4]=>1001
[10,3,1]=>8008
[10,2,2]=>6006
[10,2,1,1]=>36036
[10,1,1,1,1]=>24024
[9,5]=>2002
[9,4,1]=>20020
[9,3,2]=>40040
[9,3,1,1]=>120120
[9,2,2,1]=>180180
[9,2,1,1,1]=>480480
[9,1,1,1,1,1]=>240240
[8,6]=>3003
[8,5,1]=>36036
[8,4,2]=>90090
[8,4,1,1]=>270270
[8,3,3]=>60060
[8,3,2,1]=>1081080
[8,3,1,1,1]=>1441440
[8,2,2,2]=>270270
[8,2,2,1,1]=>3243240
[8,2,1,1,1,1]=>5405400
[8,1,1,1,1,1,1]=>2162160
[7,7]=>1716
[7,6,1]=>48048
[7,5,2]=>144144
[7,5,1,1]=>432432
[7,4,3]=>240240
[7,4,2,1]=>2162160
[7,4,1,1,1]=>2882880
[7,3,3,1]=>1441440
[7,3,2,2]=>2162160
[7,3,2,1,1]=>17297280
[7,3,1,1,1,1]=>14414400
[7,2,2,2,1]=>8648640
[7,2,2,1,1,1]=>43243200
[7,2,1,1,1,1,1]=>51891840
[7,1,1,1,1,1,1,1]=>17297280
[6,6,2]=>84084
[6,6,1,1]=>252252
[6,5,3]=>336336
[6,5,2,1]=>3027024
[6,5,1,1,1]=>4036032
[6,4,4]=>210210
[6,4,3,1]=>5045040
[6,4,2,2]=>3783780
[6,4,2,1,1]=>30270240
[6,4,1,1,1,1]=>25225200
[6,3,3,2]=>5045040
[6,3,3,1,1]=>20180160
[6,3,2,2,1]=>60540480
[6,3,2,1,1,1]=>201801600
[6,3,1,1,1,1,1]=>121080960
[6,2,2,2,2]=>7567560
[6,2,2,2,1,1]=>151351200
[6,2,2,1,1,1,1]=>454053600
[6,2,1,1,1,1,1,1]=>423783360
[6,1,1,1,1,1,1,1,1]=>121080960
[5,5,4]=>252252
[5,5,3,1]=>3027024
[5,5,2,2]=>2270268
[5,5,2,1,1]=>18162144
[5,5,1,1,1,1]=>15135120
[5,4,4,1]=>3783780
[5,4,3,2]=>15135120
[5,4,3,1,1]=>60540480
[5,4,2,2,1]=>90810720
[5,4,2,1,1,1]=>302702400
[5,4,1,1,1,1,1]=>181621440
[5,3,3,3]=>3363360
[5,3,3,2,1]=>121080960
[5,3,3,1,1,1]=>201801600
[5,3,2,2,2]=>60540480
[5,3,2,2,1,1]=>908107200
[5,3,2,1,1,1,1]=>1816214400
[5,3,1,1,1,1,1,1]=>847566720
[5,2,2,2,2,1]=>227026800
[5,2,2,2,1,1,1]=>1816214400
[5,1,1,1,1,1,1,1,1,1]=>726485760
[4,4,4,2]=>3153150
[4,4,4,1,1]=>12612600
[4,4,3,3]=>6306300
[4,4,3,2,1]=>151351200
[4,4,3,1,1,1]=>252252000
[4,4,2,2,2]=>37837800
[4,4,2,2,1,1]=>567567000
[4,4,2,1,1,1,1]=>1135134000
[4,4,1,1,1,1,1,1]=>529729200
[4,3,3,3,1]=>67267200
[4,3,3,2,2]=>151351200
[4,3,3,2,1,1]=>1513512000
[4,3,3,1,1,1,1]=>1513512000
[4,3,2,2,2,1]=>1513512000
[4,2,2,2,2,2]=>113513400
[3,3,3,3,2]=>33633600
[3,3,3,3,1,1]=>168168000
[3,3,3,2,2,1]=>1009008000
[3,3,2,2,2,2]=>378378000
[2,2,2,2,2,2,2]=>97297200
[15]=>1
[14,1]=>15
[13,2]=>105
[13,1,1]=>210
[12,3]=>455
[12,2,1]=>2730
[12,1,1,1]=>2730
[11,4]=>1365
[11,3,1]=>10920
[11,2,2]=>8190
[11,2,1,1]=>49140
[11,1,1,1,1]=>32760
[10,5]=>3003
[10,4,1]=>30030
[10,3,2]=>60060
[10,3,1,1]=>180180
[10,2,2,1]=>270270
[10,2,1,1,1]=>720720
[10,1,1,1,1,1]=>360360
[9,6]=>5005
[9,5,1]=>60060
[9,4,2]=>150150
[9,4,1,1]=>450450
[9,3,3]=>100100
[9,3,2,1]=>1801800
[9,3,1,1,1]=>2402400
[9,2,2,2]=>450450
[9,2,2,1,1]=>5405400
[9,2,1,1,1,1]=>9009000
[9,1,1,1,1,1,1]=>3603600
[8,7]=>6435
[8,6,1]=>90090
[8,5,2]=>270270
[8,5,1,1]=>810810
[8,4,3]=>450450
[8,4,2,1]=>4054050
[8,4,1,1,1]=>5405400
[8,3,3,1]=>2702700
[8,3,2,2]=>4054050
[8,3,2,1,1]=>32432400
[8,3,1,1,1,1]=>27027000
[8,2,2,2,1]=>16216200
[8,2,2,1,1,1]=>81081000
[8,2,1,1,1,1,1]=>97297200
[8,1,1,1,1,1,1,1]=>32432400
[7,7,1]=>51480
[7,6,2]=>360360
[7,6,1,1]=>1081080
[7,5,3]=>720720
[7,5,2,1]=>6486480
[7,5,1,1,1]=>8648640
[7,4,4]=>450450
[7,4,3,1]=>10810800
[7,4,2,2]=>8108100
[7,4,2,1,1]=>64864800
[7,4,1,1,1,1]=>54054000
[7,3,3,2]=>10810800
[7,3,3,1,1]=>43243200
[7,3,2,2,1]=>129729600
[7,3,2,1,1,1]=>432432000
[7,3,1,1,1,1,1]=>259459200
[7,2,2,2,2]=>16216200
[7,2,2,2,1,1]=>324324000
[7,2,2,1,1,1,1]=>972972000
[7,2,1,1,1,1,1,1]=>908107200
[7,1,1,1,1,1,1,1,1]=>259459200
[6,6,3]=>420420
[6,6,2,1]=>3783780
[6,6,1,1,1]=>5045040
[6,5,4]=>1261260
[6,5,3,1]=>15135120
[6,5,2,2]=>11351340
[6,5,2,1,1]=>90810720
[6,5,1,1,1,1]=>75675600
[6,4,4,1]=>9459450
[6,4,3,2]=>37837800
[6,4,3,1,1]=>151351200
[6,4,2,2,1]=>227026800
[6,4,2,1,1,1]=>756756000
[6,4,1,1,1,1,1]=>454053600
[6,3,3,3]=>8408400
[6,3,3,2,1]=>302702400
[6,3,3,1,1,1]=>504504000
[6,3,2,2,2]=>151351200
[6,3,1,1,1,1,1,1]=>2118916800
[6,2,2,2,2,1]=>567567000
[6,1,1,1,1,1,1,1,1,1]=>1816214400
[5,5,5]=>252252
[5,5,4,1]=>11351340
[5,5,3,2]=>22702680
[5,5,3,1,1]=>90810720
[5,5,2,2,1]=>136216080
[5,5,2,1,1,1]=>454053600
[5,5,1,1,1,1,1]=>272432160
[5,4,4,2]=>28378350
[5,4,4,1,1]=>113513400
[5,4,3,3]=>37837800
[5,4,3,2,1]=>908107200
[5,4,3,1,1,1]=>1513512000
[5,4,2,2,2]=>227026800
[5,3,3,3,1]=>201801600
[5,3,3,2,2]=>454053600
[5,2,2,2,2,2]=>340540200
[4,4,4,3]=>15765750
[4,4,4,2,1]=>189189000
[4,4,4,1,1,1]=>315315000
[4,4,3,3,1]=>378378000
[4,4,3,2,2]=>567567000
[4,3,3,3,2]=>504504000
[3,3,3,3,3]=>33633600
[16]=>1
[15,1]=>16
[14,2]=>120
[14,1,1]=>240
[13,3]=>560
[13,2,1]=>3360
[13,1,1,1]=>3360
[12,4]=>1820
[12,3,1]=>14560
[12,2,2]=>10920
[12,2,1,1]=>65520
[12,1,1,1,1]=>43680
[11,5]=>4368
[11,4,1]=>43680
[11,3,2]=>87360
[11,3,1,1]=>262080
[11,2,2,1]=>393120
[11,2,1,1,1]=>1048320
[11,1,1,1,1,1]=>524160
[10,6]=>8008
[10,5,1]=>96096
[10,4,2]=>240240
[10,4,1,1]=>720720
[10,3,3]=>160160
[10,3,2,1]=>2882880
[10,3,1,1,1]=>3843840
[10,2,2,2]=>720720
[10,2,2,1,1]=>8648640
[10,2,1,1,1,1]=>14414400
[10,1,1,1,1,1,1]=>5765760
[9,7]=>11440
[9,6,1]=>160160
[9,5,2]=>480480
[9,5,1,1]=>1441440
[9,4,3]=>800800
[9,4,2,1]=>7207200
[9,4,1,1,1]=>9609600
[9,3,3,1]=>4804800
[9,3,2,2]=>7207200
[9,3,2,1,1]=>57657600
[9,3,1,1,1,1]=>48048000
[9,2,2,2,1]=>28828800
[9,2,2,1,1,1]=>144144000
[9,2,1,1,1,1,1]=>172972800
[9,1,1,1,1,1,1,1]=>57657600
[8,8]=>6435
[8,7,1]=>205920
[8,6,2]=>720720
[8,6,1,1]=>2162160
[8,5,3]=>1441440
[8,5,2,1]=>12972960
[8,5,1,1,1]=>17297280
[8,4,4]=>900900
[8,4,3,1]=>21621600
[8,4,2,2]=>16216200
[8,4,2,1,1]=>129729600
[8,4,1,1,1,1]=>108108000
[8,3,3,2]=>21621600
[8,3,3,1,1]=>86486400
[8,3,2,2,1]=>259459200
[8,3,2,1,1,1]=>864864000
[8,3,1,1,1,1,1]=>518918400
[8,2,2,2,2]=>32432400
[8,2,2,2,1,1]=>648648000
[8,2,2,1,1,1,1]=>1945944000
[8,2,1,1,1,1,1,1]=>1816214400
[8,1,1,1,1,1,1,1,1]=>518918400
[7,7,2]=>411840
[7,7,1,1]=>1235520
[7,6,3]=>1921920
[7,6,2,1]=>17297280
[7,6,1,1,1]=>23063040
[7,5,4]=>2882880
[7,5,3,1]=>34594560
[7,5,2,2]=>25945920
[7,5,2,1,1]=>207567360
[7,5,1,1,1,1]=>172972800
[7,4,4,1]=>21621600
[7,4,3,2]=>86486400
[7,4,3,1,1]=>345945600
[7,4,2,2,1]=>518918400
[7,4,2,1,1,1]=>1729728000
[7,4,1,1,1,1,1]=>1037836800
[7,3,3,3]=>19219200
[7,3,3,2,1]=>691891200
[7,3,3,1,1,1]=>1153152000
[7,3,2,2,2]=>345945600
[7,2,2,2,2,1]=>1297296000
[6,6,4]=>1681680
[6,6,3,1]=>20180160
[6,6,2,2]=>15135120
[6,6,2,1,1]=>121080960
[6,6,1,1,1,1]=>100900800
[6,5,5]=>2018016
[6,5,4,1]=>60540480
[6,5,3,2]=>121080960
[6,5,3,1,1]=>484323840
[6,5,2,2,1]=>726485760
[6,5,1,1,1,1,1]=>1452971520
[6,4,4,2]=>75675600
[6,4,4,1,1]=>302702400
[6,4,3,3]=>100900800
[6,4,2,2,2]=>605404800
[6,3,3,3,1]=>538137600
[6,3,3,2,2]=>1210809600
[6,2,2,2,2,2]=>908107200
[5,5,5,1]=>12108096
[5,5,4,2]=>90810720
[5,5,4,1,1]=>363242880
[5,5,3,3]=>60540480
[5,5,3,2,1]=>1452971520
[5,5,2,2,2]=>363242880
[5,4,4,3]=>151351200
[5,4,4,2,1]=>1816214400
[5,3,3,3,2]=>1614412800
[4,4,4,4]=>15765750
[4,4,4,3,1]=>1009008000
[4,4,4,2,2]=>756756000
[4,3,3,3,3]=>672672000
[17]=>1
[16,1]=>17
[15,2]=>136
[15,1,1]=>272
[14,3]=>680
[14,2,1]=>4080
[14,1,1,1]=>4080
[13,4]=>2380
[13,3,1]=>19040
[13,2,2]=>14280
[13,2,1,1]=>85680
[13,1,1,1,1]=>57120
[12,5]=>6188
[12,4,1]=>61880
[12,3,2]=>123760
[12,3,1,1]=>371280
[12,2,2,1]=>556920
[12,2,1,1,1]=>1485120
[12,1,1,1,1,1]=>742560
[11,6]=>12376
[11,5,1]=>148512
[11,4,2]=>371280
[11,4,1,1]=>1113840
[11,3,3]=>247520
[11,3,2,1]=>4455360
[11,3,1,1,1]=>5940480
[11,2,2,2]=>1113840
[11,2,2,1,1]=>13366080
[11,2,1,1,1,1]=>22276800
[11,1,1,1,1,1,1]=>8910720
[10,7]=>19448
[10,6,1]=>272272
[10,5,2]=>816816
[10,5,1,1]=>2450448
[10,4,3]=>1361360
[10,4,2,1]=>12252240
[10,4,1,1,1]=>16336320
[10,3,3,1]=>8168160
[10,3,2,2]=>12252240
[10,3,2,1,1]=>98017920
[10,3,1,1,1,1]=>81681600
[10,2,2,2,1]=>49008960
[10,2,2,1,1,1]=>245044800
[10,2,1,1,1,1,1]=>294053760
[10,1,1,1,1,1,1,1]=>98017920
[9,8]=>24310
[9,7,1]=>388960
[9,6,2]=>1361360
[9,6,1,1]=>4084080
[9,5,3]=>2722720
[9,5,2,1]=>24504480
[9,5,1,1,1]=>32672640
[9,4,4]=>1701700
[9,4,3,1]=>40840800
[9,4,2,2]=>30630600
[9,4,2,1,1]=>245044800
[9,4,1,1,1,1]=>204204000
[9,3,3,2]=>40840800
[9,3,3,1,1]=>163363200
[9,3,2,2,1]=>490089600
[9,3,2,1,1,1]=>1633632000
[9,3,1,1,1,1,1]=>980179200
[9,2,2,2,2]=>61261200
[9,2,2,2,1,1]=>1225224000
[9,1,1,1,1,1,1,1,1]=>980179200
[8,8,1]=>218790
[8,7,2]=>1750320
[8,7,1,1]=>5250960
[8,6,3]=>4084080
[8,6,2,1]=>36756720
[8,6,1,1,1]=>49008960
[8,5,4]=>6126120
[8,5,3,1]=>73513440
[8,5,2,2]=>55135080
[8,5,2,1,1]=>441080640
[8,5,1,1,1,1]=>367567200
[8,4,4,1]=>45945900
[8,4,3,2]=>183783600
[8,4,3,1,1]=>735134400
[8,4,2,2,1]=>1102701600
[8,3,3,3]=>40840800
[8,3,3,2,1]=>1470268800
[8,3,2,2,2]=>735134400
[7,7,3]=>2333760
[7,7,2,1]=>21003840
[7,7,1,1,1]=>28005120
[7,6,4]=>8168160
[7,6,3,1]=>98017920
[7,6,2,2]=>73513440
[7,6,2,1,1]=>588107520
[7,6,1,1,1,1]=>490089600
[7,5,5]=>4900896
[7,5,4,1]=>147026880
[7,5,3,2]=>294053760
[7,5,3,1,1]=>1176215040
[7,5,2,2,1]=>1764322560
[7,4,4,2]=>183783600
[7,4,4,1,1]=>735134400
[7,4,3,3]=>245044800
[7,4,2,2,2]=>1470268800
[7,3,3,3,1]=>1306905600
[6,6,5]=>5717712
[6,6,4,1]=>85765680
[6,6,3,2]=>171531360
[6,6,3,1,1]=>686125440
[6,6,2,2,1]=>1029188160
[6,6,1,1,1,1,1]=>2058376320
[6,5,5,1]=>102918816
[6,5,4,2]=>514594080
[6,5,4,1,1]=>2058376320
[6,5,3,3]=>343062720
[6,5,2,2,2]=>2058376320
[6,4,4,3]=>428828400
[5,5,5,2]=>102918816
[5,5,5,1,1]=>411675264
[5,5,4,3]=>514594080
[5,4,4,4]=>214414200
[4,4,4,4,1]=>1072071000
                    

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Description

The coefficient of the monomial corresponding to the integer partition in a certain power series.
Let $f(x) = 1 - c_1 x/1! - c_2 x^2/2! - \dots$ and let $F(x) = f'(x)/f(x) = 1 + C_1 x/1! + C_2 x^2/2! + \dots$.
Then $C_{n-1}$ is a polynomial in $c_1,\dots,c_n$, whose monomials are indexed by integer partitions of $n$. This statistic records its coefficients.

References

[1] Irregular triangle read by rows: row n gives coefficients of n-th logarithmic polynomial L_n(x_1, x_2, ...) with monomials sorted into standard order. OEIS:A263634

Code

def statistic(pi):
    R. = InfinitePolynomialRing(QQ)
    L. = LazyPowerSeriesRing(R)
    f = L(lambda n: -a[n]/factorial(n) if n else 1)
    pi = Partition(pi)
    n = sum(pi)
    P = PolynomialRing(QQ, [f"a_{i}" for i in range(1, n+2)])
    p = P(-diff(log(f))[n-1])*factorial(n-1)
    return p.monomial_coefficient(P.monomial(tuple(pi.to_exp(n+1))))