***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001387 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: Number of standard Young tableaux of the skew shape tracing the border of the given partition. Let $\lambda \vdash n$ be a diagram with the given partition as shape. Add $n$ additional boxes, one in each column $1,\dotsc,n$, and let this be $\mu$. The statistic is the number of standard Young tableaux of skew shape $\mu/\lambda$, which is equal to $\frac{n!}{\prod_{i} (\mu_i - \lambda_i)!}$. For example, $\lambda=[2,1,1]$ gives $\mu = [4,2,1,1]$. The first row in the skew shape $\mu/\lambda$ has two boxes, so the number of SYT of shape $\mu/\lambda$ is then $4!/2 = 12$. This statistic shows up in the study of skew specialized Macdonald polynomials, where a type of charge statistic give rise to a $q$-analogue of the above formula. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(la): la = Partition(la) n = la.size() la_t = la.conjugate() mu = Partition([c + 1 for c in la_t] + [1]*(n-len(la_t))) return StandardTableaux(SkewPartition([mu.conjugate(), la])).cardinality() ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 6 [1,1,1] => 3 [4] => 1 [3,1] => 12 [2,2] => 6 [2,1,1] => 12 [1,1,1,1] => 4 [5] => 1 [4,1] => 20 [3,2] => 30 [3,1,1] => 30 [2,2,1] => 20 [2,1,1,1] => 20 [1,1,1,1,1] => 5 [6] => 1 [5,1] => 30 [4,2] => 90 [4,1,1] => 60 [3,3] => 20 [3,2,1] => 120 [3,1,1,1] => 60 [2,2,2] => 15 [2,2,1,1] => 30 [2,1,1,1,1] => 30 [1,1,1,1,1,1] => 6 [7] => 1 [6,1] => 42 [5,2] => 210 [5,1,1] => 105 [4,3] => 140 [4,2,1] => 420 [4,1,1,1] => 140 [3,3,1] => 105 [3,2,2] => 105 [3,2,1,1] => 210 [3,1,1,1,1] => 105 [2,2,2,1] => 42 [2,2,1,1,1] => 42 [2,1,1,1,1,1] => 42 [1,1,1,1,1,1,1] => 7 [8] => 1 [7,1] => 56 [6,2] => 420 [6,1,1] => 168 [5,3] => 560 [5,2,1] => 1120 [5,1,1,1] => 280 [4,4] => 70 [4,3,1] => 840 [4,2,2] => 420 [4,2,1,1] => 840 [4,1,1,1,1] => 280 [3,3,2] => 168 [3,3,1,1] => 168 [3,2,2,1] => 336 [3,2,1,1,1] => 336 [3,1,1,1,1,1] => 168 [2,2,2,2] => 28 [2,2,2,1,1] => 56 [2,2,1,1,1,1] => 56 [2,1,1,1,1,1,1] => 56 [1,1,1,1,1,1,1,1] => 8 [9] => 1 [8,1] => 72 [7,2] => 756 [7,1,1] => 252 [6,3] => 1680 [6,2,1] => 2520 [6,1,1,1] => 504 [5,4] => 630 [5,3,1] => 3780 [5,2,2] => 1260 [5,2,1,1] => 2520 [5,1,1,1,1] => 630 [4,4,1] => 504 [4,3,2] => 1512 [4,3,1,1] => 1512 [4,2,2,1] => 1512 [4,2,1,1,1] => 1512 [4,1,1,1,1,1] => 504 [3,3,3] => 84 [3,3,2,1] => 504 [3,3,1,1,1] => 252 [3,2,2,2] => 252 [3,2,2,1,1] => 504 [3,2,1,1,1,1] => 504 [3,1,1,1,1,1,1] => 252 [2,2,2,2,1] => 72 [2,2,2,1,1,1] => 72 [2,2,1,1,1,1,1] => 72 [2,1,1,1,1,1,1,1] => 72 [1,1,1,1,1,1,1,1,1] => 9 [10] => 1 [9,1] => 90 [8,2] => 1260 [8,1,1] => 360 [7,3] => 4200 [7,2,1] => 5040 [7,1,1,1] => 840 [6,4] => 3150 [6,3,1] => 12600 [6,2,2] => 3150 [6,2,1,1] => 6300 [6,1,1,1,1] => 1260 [5,5] => 252 [5,4,1] => 5040 [5,3,2] => 7560 [5,3,1,1] => 7560 [5,2,2,1] => 5040 [5,2,1,1,1] => 5040 [5,1,1,1,1,1] => 1260 [4,4,2] => 1260 [4,4,1,1] => 840 [4,3,3] => 840 [4,3,2,1] => 5040 [4,3,1,1,1] => 2520 [4,2,2,2] => 1260 [4,2,2,1,1] => 2520 [4,2,1,1,1,1] => 2520 [4,1,1,1,1,1,1] => 840 [3,3,3,1] => 360 [3,3,2,2] => 360 [3,3,2,1,1] => 720 [3,3,1,1,1,1] => 360 [3,2,2,2,1] => 720 [3,2,2,1,1,1] => 720 [3,2,1,1,1,1,1] => 720 [3,1,1,1,1,1,1,1] => 360 [2,2,2,2,2] => 45 [2,2,2,2,1,1] => 90 [2,2,2,1,1,1,1] => 90 [2,2,1,1,1,1,1,1] => 90 [2,1,1,1,1,1,1,1,1] => 90 [1,1,1,1,1,1,1,1,1,1] => 10 [11] => 1 [10,1] => 110 [9,2] => 1980 [9,1,1] => 495 [8,3] => 9240 [8,2,1] => 9240 [8,1,1,1] => 1320 [7,4] => 11550 [7,3,1] => 34650 [7,2,2] => 6930 [7,2,1,1] => 13860 [7,1,1,1,1] => 2310 [6,5] => 2772 [6,4,1] => 27720 [6,3,2] => 27720 [6,3,1,1] => 27720 [6,2,2,1] => 13860 [6,2,1,1,1] => 13860 [6,1,1,1,1,1] => 2772 [5,5,1] => 2310 [5,4,2] => 13860 [5,4,1,1] => 9240 [5,3,3] => 4620 [5,3,2,1] => 27720 [5,3,1,1,1] => 13860 [5,2,2,2] => 4620 [5,2,2,1,1] => 9240 [5,2,1,1,1,1] => 9240 [5,1,1,1,1,1,1] => 2310 [4,4,3] => 1320 [4,4,2,1] => 3960 [4,4,1,1,1] => 1320 [4,3,3,1] => 3960 [4,3,2,2] => 3960 [4,3,2,1,1] => 7920 [4,3,1,1,1,1] => 3960 [4,2,2,2,1] => 3960 [4,2,2,1,1,1] => 3960 [4,2,1,1,1,1,1] => 3960 [4,1,1,1,1,1,1,1] => 1320 [3,3,3,2] => 495 [3,3,3,1,1] => 495 [3,3,2,2,1] => 990 [3,3,2,1,1,1] => 990 [3,3,1,1,1,1,1] => 495 [3,2,2,2,2] => 495 [3,2,2,2,1,1] => 990 [3,2,2,1,1,1,1] => 990 [3,2,1,1,1,1,1,1] => 990 [3,1,1,1,1,1,1,1,1] => 495 [2,2,2,2,2,1] => 110 [2,2,2,2,1,1,1] => 110 [2,2,2,1,1,1,1,1] => 110 [2,2,1,1,1,1,1,1,1] => 110 [2,1,1,1,1,1,1,1,1,1] => 110 [1,1,1,1,1,1,1,1,1,1,1] => 11 [12] => 1 [11,1] => 132 [10,2] => 2970 [10,1,1] => 660 [9,3] => 18480 [9,2,1] => 15840 [9,1,1,1] => 1980 [8,4] => 34650 [8,3,1] => 83160 [8,2,2] => 13860 [8,2,1,1] => 27720 [8,1,1,1,1] => 3960 [7,5] => 16632 [7,4,1] => 110880 [7,3,2] => 83160 [7,3,1,1] => 83160 [7,2,2,1] => 33264 [7,2,1,1,1] => 33264 [7,1,1,1,1,1] => 5544 [6,6] => 924 [6,5,1] => 27720 [6,4,2] => 83160 [6,4,1,1] => 55440 [6,3,3] => 18480 [6,3,2,1] => 110880 [6,3,1,1,1] => 55440 [6,2,2,2] => 13860 [6,2,2,1,1] => 27720 [6,2,1,1,1,1] => 27720 [6,1,1,1,1,1,1] => 5544 [5,5,2] => 7920 [5,5,1,1] => 3960 [5,4,3] => 15840 [5,4,2,1] => 47520 [5,4,1,1,1] => 15840 [5,3,3,1] => 23760 [5,3,2,2] => 23760 [5,3,2,1,1] => 47520 [5,3,1,1,1,1] => 23760 [5,2,2,2,1] => 15840 [5,2,2,1,1,1] => 15840 [5,2,1,1,1,1,1] => 15840 [5,1,1,1,1,1,1,1] => 3960 [4,4,4] => 495 [4,4,3,1] => 5940 [4,4,2,2] => 2970 [4,4,2,1,1] => 5940 [4,4,1,1,1,1] => 1980 [4,3,3,2] => 5940 [4,3,3,1,1] => 5940 [4,3,2,2,1] => 11880 [4,3,2,1,1,1] => 11880 [4,3,1,1,1,1,1] => 5940 [4,2,2,2,2] => 2970 [4,2,2,2,1,1] => 5940 [4,2,2,1,1,1,1] => 5940 [4,2,1,1,1,1,1,1] => 5940 [4,1,1,1,1,1,1,1,1] => 1980 [3,3,3,3] => 220 [3,3,3,2,1] => 1320 [3,3,3,1,1,1] => 660 [3,3,2,2,2] => 660 [3,3,2,2,1,1] => 1320 [3,3,2,1,1,1,1] => 1320 [3,3,1,1,1,1,1,1] => 660 [3,2,2,2,2,1] => 1320 [3,2,2,2,1,1,1] => 1320 [3,2,2,1,1,1,1,1] => 1320 [3,2,1,1,1,1,1,1,1] => 1320 [3,1,1,1,1,1,1,1,1,1] => 660 [2,2,2,2,2,2] => 66 [2,2,2,2,2,1,1] => 132 [2,2,2,2,1,1,1,1] => 132 [2,2,2,1,1,1,1,1,1] => 132 [2,2,1,1,1,1,1,1,1,1] => 132 [2,1,1,1,1,1,1,1,1,1,1] => 132 [1,1,1,1,1,1,1,1,1,1,1,1] => 12 [13] => 1 [12,1] => 156 [11,2] => 4290 [11,1,1] => 858 [10,3] => 34320 [10,2,1] => 25740 [10,1,1,1] => 2860 [9,4] => 90090 [9,3,1] => 180180 [9,2,2] => 25740 [9,2,1,1] => 51480 [9,1,1,1,1] => 6435 [8,5] => 72072 [8,4,1] => 360360 [8,3,2] => 216216 [8,3,1,1] => 216216 [8,2,2,1] => 72072 [8,2,1,1,1] => 72072 [8,1,1,1,1,1] => 10296 [7,6] => 12012 [7,5,1] => 180180 [7,4,2] => 360360 [7,4,1,1] => 240240 [7,3,3] => 60060 [7,3,2,1] => 360360 [7,3,1,1,1] => 180180 [7,2,2,2] => 36036 [7,2,2,1,1] => 72072 [7,2,1,1,1,1] => 72072 [7,1,1,1,1,1,1] => 12012 [6,6,1] => 10296 [6,5,2] => 102960 [6,5,1,1] => 51480 [6,4,3] => 102960 [6,4,2,1] => 308880 [6,4,1,1,1] => 102960 [6,3,3,1] => 102960 [6,3,2,2] => 102960 [6,3,2,1,1] => 205920 [6,3,1,1,1,1] => 102960 [6,2,2,2,1] => 51480 [6,2,2,1,1,1] => 51480 [6,2,1,1,1,1,1] => 51480 [6,1,1,1,1,1,1,1] => 10296 [5,5,3] => 12870 [5,5,2,1] => 25740 [5,5,1,1,1] => 6435 [5,4,4] => 6435 [5,4,3,1] => 77220 [5,4,2,2] => 38610 [5,4,2,1,1] => 77220 [5,4,1,1,1,1] => 25740 [5,3,3,2] => 38610 [5,3,3,1,1] => 38610 [5,3,2,2,1] => 77220 [5,3,2,1,1,1] => 77220 [5,3,1,1,1,1,1] => 38610 [5,2,2,2,2] => 12870 [5,2,2,2,1,1] => 25740 [5,2,2,1,1,1,1] => 25740 [5,2,1,1,1,1,1,1] => 25740 [5,1,1,1,1,1,1,1,1] => 6435 [4,4,4,1] => 2860 [4,4,3,2] => 8580 [4,4,3,1,1] => 8580 [4,4,2,2,1] => 8580 [4,4,2,1,1,1] => 8580 [4,4,1,1,1,1,1] => 2860 [4,3,3,3] => 2860 [4,3,3,2,1] => 17160 [4,3,3,1,1,1] => 8580 [4,3,2,2,2] => 8580 [4,3,2,2,1,1] => 17160 [4,3,2,1,1,1,1] => 17160 [4,3,1,1,1,1,1,1] => 8580 [4,2,2,2,2,1] => 8580 [4,2,2,2,1,1,1] => 8580 [4,2,2,1,1,1,1,1] => 8580 [4,2,1,1,1,1,1,1,1] => 8580 [4,1,1,1,1,1,1,1,1,1] => 2860 [3,3,3,3,1] => 858 [3,3,3,2,2] => 858 [3,3,3,2,1,1] => 1716 [3,3,3,1,1,1,1] => 858 [3,3,2,2,2,1] => 1716 [3,3,2,2,1,1,1] => 1716 [3,3,2,1,1,1,1,1] => 1716 [3,3,1,1,1,1,1,1,1] => 858 [3,2,2,2,2,2] => 858 [3,2,2,2,2,1,1] => 1716 [3,2,2,2,1,1,1,1] => 1716 [3,2,2,1,1,1,1,1,1] => 1716 [3,2,1,1,1,1,1,1,1,1] => 1716 [3,1,1,1,1,1,1,1,1,1,1] => 858 [2,2,2,2,2,2,1] => 156 [2,2,2,2,2,1,1,1] => 156 [2,2,2,2,1,1,1,1,1] => 156 [2,2,2,1,1,1,1,1,1,1] => 156 [2,2,1,1,1,1,1,1,1,1,1] => 156 [2,1,1,1,1,1,1,1,1,1,1,1] => 156 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 13 [14] => 1 [13,1] => 182 [12,2] => 6006 [12,1,1] => 1092 [11,3] => 60060 [11,2,1] => 40040 [11,1,1,1] => 4004 [10,4] => 210210 [10,3,1] => 360360 [10,2,2] => 45045 [10,2,1,1] => 90090 [10,1,1,1,1] => 10010 [9,5] => 252252 [9,4,1] => 1009008 [9,3,2] => 504504 [9,3,1,1] => 504504 [9,2,2,1] => 144144 [9,2,1,1,1] => 144144 [9,1,1,1,1,1] => 18018 [8,6] => 84084 [8,5,1] => 840840 [8,4,2] => 1261260 [8,4,1,1] => 840840 [8,3,3] => 168168 [8,3,2,1] => 1009008 [8,3,1,1,1] => 504504 [8,2,2,2] => 84084 [8,2,2,1,1] => 168168 [8,2,1,1,1,1] => 168168 [8,1,1,1,1,1,1] => 24024 [7,7] => 3432 [7,6,1] => 144144 [7,5,2] => 720720 [7,5,1,1] => 360360 [7,4,3] => 480480 [7,4,2,1] => 1441440 [7,4,1,1,1] => 480480 [7,3,3,1] => 360360 [7,3,2,2] => 360360 [7,3,2,1,1] => 720720 [7,3,1,1,1,1] => 360360 [7,2,2,2,1] => 144144 [7,2,2,1,1,1] => 144144 [7,2,1,1,1,1,1] => 144144 [7,1,1,1,1,1,1,1] => 24024 [6,6,2] => 45045 [6,6,1,1] => 18018 [6,5,3] => 180180 [6,5,2,1] => 360360 [6,5,1,1,1] => 90090 [6,4,4] => 45045 [6,4,3,1] => 540540 [6,4,2,2] => 270270 [6,4,2,1,1] => 540540 [6,4,1,1,1,1] => 180180 [6,3,3,2] => 180180 [6,3,3,1,1] => 180180 [6,3,2,2,1] => 360360 [6,3,2,1,1,1] => 360360 [6,3,1,1,1,1,1] => 180180 [6,2,2,2,2] => 45045 [6,2,2,2,1,1] => 90090 [6,2,2,1,1,1,1] => 90090 [6,2,1,1,1,1,1,1] => 90090 [6,1,1,1,1,1,1,1,1] => 18018 [5,5,4] => 10010 [5,5,3,1] => 60060 [5,5,2,2] => 20020 [5,5,2,1,1] => 40040 [5,5,1,1,1,1] => 10010 [5,4,4,1] => 40040 [5,4,3,2] => 120120 [5,4,3,1,1] => 120120 [5,4,2,2,1] => 120120 [5,4,2,1,1,1] => 120120 [5,4,1,1,1,1,1] => 40040 [5,3,3,3] => 20020 [5,3,3,2,1] => 120120 [5,3,3,1,1,1] => 60060 [5,3,2,2,2] => 60060 [5,3,2,2,1,1] => 120120 [5,3,2,1,1,1,1] => 120120 [5,3,1,1,1,1,1,1] => 60060 [5,2,2,2,2,1] => 40040 [5,2,2,2,1,1,1] => 40040 [5,2,2,1,1,1,1,1] => 40040 [5,2,1,1,1,1,1,1,1] => 40040 [5,1,1,1,1,1,1,1,1,1] => 10010 [4,4,4,2] => 6006 [4,4,4,1,1] => 4004 [4,4,3,3] => 4004 [4,4,3,2,1] => 24024 [4,4,3,1,1,1] => 12012 [4,4,2,2,2] => 6006 [4,4,2,2,1,1] => 12012 [4,4,2,1,1,1,1] => 12012 [4,4,1,1,1,1,1,1] => 4004 [4,3,3,3,1] => 12012 [4,3,3,2,2] => 12012 [4,3,3,2,1,1] => 24024 [4,3,3,1,1,1,1] => 12012 [4,3,2,2,2,1] => 24024 [4,3,2,2,1,1,1] => 24024 [4,3,2,1,1,1,1,1] => 24024 [4,3,1,1,1,1,1,1,1] => 12012 [4,2,2,2,2,2] => 6006 [4,2,2,2,2,1,1] => 12012 [4,2,2,2,1,1,1,1] => 12012 [4,2,2,1,1,1,1,1,1] => 12012 [4,2,1,1,1,1,1,1,1,1] => 12012 [4,1,1,1,1,1,1,1,1,1,1] => 4004 [3,3,3,3,2] => 1092 [3,3,3,3,1,1] => 1092 [3,3,3,2,2,1] => 2184 [3,3,3,2,1,1,1] => 2184 [3,3,3,1,1,1,1,1] => 1092 [3,3,2,2,2,2] => 1092 [3,3,2,2,2,1,1] => 2184 [3,3,2,2,1,1,1,1] => 2184 [3,3,2,1,1,1,1,1,1] => 2184 [3,3,1,1,1,1,1,1,1,1] => 1092 [3,2,2,2,2,2,1] => 2184 [3,2,2,2,2,1,1,1] => 2184 [3,2,2,2,1,1,1,1,1] => 2184 [3,2,2,1,1,1,1,1,1,1] => 2184 [3,2,1,1,1,1,1,1,1,1,1] => 2184 [3,1,1,1,1,1,1,1,1,1,1,1] => 1092 [2,2,2,2,2,2,2] => 91 [2,2,2,2,2,2,1,1] => 182 [2,2,2,2,2,1,1,1,1] => 182 [2,2,2,2,1,1,1,1,1,1] => 182 [2,2,2,1,1,1,1,1,1,1,1] => 182 [2,2,1,1,1,1,1,1,1,1,1,1] => 182 [2,1,1,1,1,1,1,1,1,1,1,1,1] => 182 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14 [15] => 1 [14,1] => 210 [13,2] => 8190 [13,1,1] => 1365 [12,3] => 100100 [12,2,1] => 60060 [12,1,1,1] => 5460 [11,4] => 450450 [11,3,1] => 675675 [11,2,2] => 75075 [11,2,1,1] => 150150 [11,1,1,1,1] => 15015 [10,5] => 756756 [10,4,1] => 2522520 [10,3,2] => 1081080 [10,3,1,1] => 1081080 [10,2,2,1] => 270270 [10,2,1,1,1] => 270270 [10,1,1,1,1,1] => 30030 [9,6] => 420420 [9,5,1] => 3153150 [9,4,2] => 3783780 [9,4,1,1] => 2522520 [9,3,3] => 420420 [9,3,2,1] => 2522520 [9,3,1,1,1] => 1261260 [9,2,2,2] => 180180 [9,2,2,1,1] => 360360 [9,2,1,1,1,1] => 360360 [9,1,1,1,1,1,1] => 45045 [8,7] => 51480 [8,6,1] => 1081080 [8,5,2] => 3603600 [8,5,1,1] => 1801800 [8,4,3] => 1801800 [8,4,2,1] => 5405400 [8,4,1,1,1] => 1801800 [8,3,3,1] => 1081080 [8,3,2,2] => 1081080 [8,3,2,1,1] => 2162160 [8,3,1,1,1,1] => 1081080 [8,2,2,2,1] => 360360 [8,2,2,1,1,1] => 360360 [8,2,1,1,1,1,1] => 360360 [8,1,1,1,1,1,1,1] => 51480 [7,7,1] => 45045 [7,6,2] => 675675 [7,6,1,1] => 270270 [7,5,3] => 1351350 [7,5,2,1] => 2702700 [7,5,1,1,1] => 675675 [7,4,4] => 225225 [7,4,3,1] => 2702700 [7,4,2,2] => 1351350 [7,4,2,1,1] => 2702700 [7,4,1,1,1,1] => 900900 [7,3,3,2] => 675675 [7,3,3,1,1] => 675675 [7,3,2,2,1] => 1351350 [7,3,2,1,1,1] => 1351350 [7,3,1,1,1,1,1] => 675675 [7,2,2,2,2] => 135135 [7,2,2,2,1,1] => 270270 [7,2,2,1,1,1,1] => 270270 [7,2,1,1,1,1,1,1] => 270270 [7,1,1,1,1,1,1,1,1] => 45045 [6,6,3] => 100100 [6,6,2,1] => 150150 [6,6,1,1,1] => 30030 [6,5,4] => 150150 [6,5,3,1] => 900900 [6,5,2,2] => 300300 [6,5,2,1,1] => 600600 [6,5,1,1,1,1] => 150150 [6,4,4,1] => 300300 [6,4,3,2] => 900900 [6,4,3,1,1] => 900900 [6,4,2,2,1] => 900900 [6,4,2,1,1,1] => 900900 [6,4,1,1,1,1,1] => 300300 [6,3,3,3] => 100100 [6,3,3,2,1] => 600600 [6,3,3,1,1,1] => 300300 [6,3,2,2,2] => 300300 [6,3,2,2,1,1] => 600600 [6,3,2,1,1,1,1] => 600600 [6,3,1,1,1,1,1,1] => 300300 [6,2,2,2,2,1] => 150150 [6,2,2,2,1,1,1] => 150150 [6,2,2,1,1,1,1,1] => 150150 [6,2,1,1,1,1,1,1,1] => 150150 [6,1,1,1,1,1,1,1,1,1] => 30030 [5,5,5] => 3003 [5,5,4,1] => 60060 [5,5,3,2] => 90090 [5,5,3,1,1] => 90090 [5,5,2,2,1] => 60060 [5,5,2,1,1,1] => 60060 [5,5,1,1,1,1,1] => 15015 [5,4,4,2] => 90090 [5,4,4,1,1] => 60060 [5,4,3,3] => 60060 [5,4,3,2,1] => 360360 [5,4,3,1,1,1] => 180180 [5,4,2,2,2] => 90090 [5,4,2,2,1,1] => 180180 [5,4,2,1,1,1,1] => 180180 [5,4,1,1,1,1,1,1] => 60060 [5,3,3,3,1] => 90090 [5,3,3,2,2] => 90090 [5,3,3,2,1,1] => 180180 [5,3,3,1,1,1,1] => 90090 [5,3,2,2,2,1] => 180180 [5,3,2,2,1,1,1] => 180180 [5,3,2,1,1,1,1,1] => 180180 [5,3,1,1,1,1,1,1,1] => 90090 [5,2,2,2,2,2] => 30030 [5,2,2,2,2,1,1] => 60060 [5,2,2,2,1,1,1,1] => 60060 [5,2,2,1,1,1,1,1,1] => 60060 [5,2,1,1,1,1,1,1,1,1] => 60060 [5,1,1,1,1,1,1,1,1,1,1] => 15015 [4,4,4,3] => 5460 [4,4,4,2,1] => 16380 [4,4,4,1,1,1] => 5460 [4,4,3,3,1] => 16380 [4,4,3,2,2] => 16380 [4,4,3,2,1,1] => 32760 [4,4,3,1,1,1,1] => 16380 [4,4,2,2,2,1] => 16380 [4,4,2,2,1,1,1] => 16380 [4,4,2,1,1,1,1,1] => 16380 [4,4,1,1,1,1,1,1,1] => 5460 [4,3,3,3,2] => 16380 [4,3,3,3,1,1] => 16380 [4,3,3,2,2,1] => 32760 [4,3,3,2,1,1,1] => 32760 [4,3,3,1,1,1,1,1] => 16380 [4,3,2,2,2,2] => 16380 [4,3,2,2,2,1,1] => 32760 [4,3,2,2,1,1,1,1] => 32760 [4,3,2,1,1,1,1,1,1] => 32760 [4,3,1,1,1,1,1,1,1,1] => 16380 [4,2,2,2,2,2,1] => 16380 [4,2,2,2,2,1,1,1] => 16380 [4,2,2,2,1,1,1,1,1] => 16380 [4,2,2,1,1,1,1,1,1,1] => 16380 [4,2,1,1,1,1,1,1,1,1,1] => 16380 [4,1,1,1,1,1,1,1,1,1,1,1] => 5460 [3,3,3,3,3] => 455 [3,3,3,3,2,1] => 2730 [3,3,3,3,1,1,1] => 1365 [3,3,3,2,2,2] => 1365 [3,3,3,2,2,1,1] => 2730 [3,3,3,2,1,1,1,1] => 2730 [3,3,3,1,1,1,1,1,1] => 1365 [3,3,2,2,2,2,1] => 2730 [3,3,2,2,2,1,1,1] => 2730 [3,3,2,2,1,1,1,1,1] => 2730 [3,3,2,1,1,1,1,1,1,1] => 2730 [3,3,1,1,1,1,1,1,1,1,1] => 1365 [3,2,2,2,2,2,2] => 1365 [3,2,2,2,2,2,1,1] => 2730 [3,2,2,2,2,1,1,1,1] => 2730 [3,2,2,2,1,1,1,1,1,1] => 2730 [3,2,2,1,1,1,1,1,1,1,1] => 2730 [3,2,1,1,1,1,1,1,1,1,1,1] => 2730 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 1365 [2,2,2,2,2,2,2,1] => 210 [2,2,2,2,2,2,1,1,1] => 210 [2,2,2,2,2,1,1,1,1,1] => 210 [2,2,2,2,1,1,1,1,1,1,1] => 210 [2,2,2,1,1,1,1,1,1,1,1,1] => 210 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 210 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 210 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15 [16] => 1 [15,1] => 240 [14,2] => 10920 [14,1,1] => 1680 [13,3] => 160160 [13,2,1] => 87360 [13,1,1,1] => 7280 [12,4] => 900900 [12,3,1] => 1201200 [12,2,2] => 120120 [12,2,1,1] => 240240 [12,1,1,1,1] => 21840 [11,5] => 2018016 [11,4,1] => 5765760 [11,3,2] => 2162160 [11,3,1,1] => 2162160 [11,2,2,1] => 480480 [11,2,1,1,1] => 480480 [11,1,1,1,1,1] => 48048 [10,6] => 1681680 [10,5,1] => 10090080 [10,4,2] => 10090080 [10,4,1,1] => 6726720 [10,3,3] => 960960 [10,3,2,1] => 5765760 [10,3,1,1,1] => 2882880 [10,2,2,2] => 360360 [10,2,2,1,1] => 720720 [10,2,1,1,1,1] => 720720 [10,1,1,1,1,1,1] => 80080 [9,7] => 411840 [9,6,1] => 5765760 [9,5,2] => 14414400 [9,5,1,1] => 7207200 [9,4,3] => 5765760 [9,4,2,1] => 17297280 [9,4,1,1,1] => 5765760 [9,3,3,1] => 2882880 [9,3,2,2] => 2882880 [9,3,2,1,1] => 5765760 [9,3,1,1,1,1] => 2882880 [9,2,2,2,1] => 823680 [9,2,2,1,1,1] => 823680 [9,2,1,1,1,1,1] => 823680 [9,1,1,1,1,1,1,1] => 102960 [8,8] => 12870 [8,7,1] => 720720 [8,6,2] => 5405400 [8,6,1,1] => 2162160 [8,5,3] => 7207200 [8,5,2,1] => 14414400 [8,5,1,1,1] => 3603600 [8,4,4] => 900900 [8,4,3,1] => 10810800 [8,4,2,2] => 5405400 [8,4,2,1,1] => 10810800 [8,4,1,1,1,1] => 3603600 [8,3,3,2] => 2162160 [8,3,3,1,1] => 2162160 [8,3,2,2,1] => 4324320 [8,3,2,1,1,1] => 4324320 [8,3,1,1,1,1,1] => 2162160 [8,2,2,2,2] => 360360 [8,2,2,2,1,1] => 720720 [8,2,2,1,1,1,1] => 720720 [8,2,1,1,1,1,1,1] => 720720 [8,1,1,1,1,1,1,1,1] => 102960 [7,7,2] => 240240 [7,7,1,1] => 80080 [7,6,3] => 1601600 [7,6,2,1] => 2402400 [7,6,1,1,1] => 480480 [7,5,4] => 1201200 [7,5,3,1] => 7207200 [7,5,2,2] => 2402400 [7,5,2,1,1] => 4804800 [7,5,1,1,1,1] => 1201200 [7,4,4,1] => 1601600 [7,4,3,2] => 4804800 [7,4,3,1,1] => 4804800 [7,4,2,2,1] => 4804800 [7,4,2,1,1,1] => 4804800 [7,4,1,1,1,1,1] => 1601600 [7,3,3,3] => 400400 [7,3,3,2,1] => 2402400 [7,3,3,1,1,1] => 1201200 [7,3,2,2,2] => 1201200 [7,3,2,2,1,1] => 2402400 [7,3,2,1,1,1,1] => 2402400 [7,3,1,1,1,1,1,1] => 1201200 [7,2,2,2,2,1] => 480480 [7,2,2,2,1,1,1] => 480480 [7,2,2,1,1,1,1,1] => 480480 [7,2,1,1,1,1,1,1,1] => 480480 [7,1,1,1,1,1,1,1,1,1] => 80080 [6,6,4] => 120120 [6,6,3,1] => 480480 [6,6,2,2] => 120120 [6,6,2,1,1] => 240240 [6,6,1,1,1,1] => 48048 [6,5,5] => 48048 [6,5,4,1] => 960960 [6,5,3,2] => 1441440 [6,5,3,1,1] => 1441440 [6,5,2,2,1] => 960960 [6,5,2,1,1,1] => 960960 [6,5,1,1,1,1,1] => 240240 [6,4,4,2] => 720720 [6,4,4,1,1] => 480480 [6,4,3,3] => 480480 [6,4,3,2,1] => 2882880 [6,4,3,1,1,1] => 1441440 [6,4,2,2,2] => 720720 [6,4,2,2,1,1] => 1441440 [6,4,2,1,1,1,1] => 1441440 [6,4,1,1,1,1,1,1] => 480480 [6,3,3,3,1] => 480480 [6,3,3,2,2] => 480480 [6,3,3,2,1,1] => 960960 [6,3,3,1,1,1,1] => 480480 [6,3,2,2,2,1] => 960960 [6,3,2,2,1,1,1] => 960960 [6,3,2,1,1,1,1,1] => 960960 [6,3,1,1,1,1,1,1,1] => 480480 [6,2,2,2,2,2] => 120120 [6,2,2,2,2,1,1] => 240240 [6,2,2,2,1,1,1,1] => 240240 [6,2,2,1,1,1,1,1,1] => 240240 [6,2,1,1,1,1,1,1,1,1] => 240240 [6,1,1,1,1,1,1,1,1,1,1] => 48048 [5,5,5,1] => 21840 [5,5,4,2] => 131040 [5,5,4,1,1] => 87360 [5,5,3,3] => 43680 [5,5,3,2,1] => 262080 [5,5,3,1,1,1] => 131040 [5,5,2,2,2] => 43680 [5,5,2,2,1,1] => 87360 [5,5,2,1,1,1,1] => 87360 [5,5,1,1,1,1,1,1] => 21840 [5,4,4,3] => 87360 [5,4,4,2,1] => 262080 [5,4,4,1,1,1] => 87360 [5,4,3,3,1] => 262080 [5,4,3,2,2] => 262080 [5,4,3,2,1,1] => 524160 [5,4,3,1,1,1,1] => 262080 [5,4,2,2,2,1] => 262080 [5,4,2,2,1,1,1] => 262080 [5,4,2,1,1,1,1,1] => 262080 [5,4,1,1,1,1,1,1,1] => 87360 [5,3,3,3,2] => 131040 [5,3,3,3,1,1] => 131040 [5,3,3,2,2,1] => 262080 [5,3,3,2,1,1,1] => 262080 [5,3,3,1,1,1,1,1] => 131040 [5,3,2,2,2,2] => 131040 [5,3,2,2,2,1,1] => 262080 [5,3,2,2,1,1,1,1] => 262080 [5,3,2,1,1,1,1,1,1] => 262080 [5,3,1,1,1,1,1,1,1,1] => 131040 [5,2,2,2,2,2,1] => 87360 [5,2,2,2,2,1,1,1] => 87360 [5,2,2,2,1,1,1,1,1] => 87360 [5,2,2,1,1,1,1,1,1,1] => 87360 [5,2,1,1,1,1,1,1,1,1,1] => 87360 [5,1,1,1,1,1,1,1,1,1,1,1] => 21840 [4,4,4,4] => 1820 [4,4,4,3,1] => 21840 [4,4,4,2,2] => 10920 [4,4,4,2,1,1] => 21840 [4,4,4,1,1,1,1] => 7280 [4,4,3,3,2] => 21840 [4,4,3,3,1,1] => 21840 [4,4,3,2,2,1] => 43680 [4,4,3,2,1,1,1] => 43680 [4,4,3,1,1,1,1,1] => 21840 [4,4,2,2,2,2] => 10920 [4,4,2,2,2,1,1] => 21840 [4,4,2,2,1,1,1,1] => 21840 [4,4,2,1,1,1,1,1,1] => 21840 [4,4,1,1,1,1,1,1,1,1] => 7280 [4,3,3,3,3] => 7280 [4,3,3,3,2,1] => 43680 [4,3,3,3,1,1,1] => 21840 [4,3,3,2,2,2] => 21840 [4,3,3,2,2,1,1] => 43680 [4,3,3,2,1,1,1,1] => 43680 [4,3,3,1,1,1,1,1,1] => 21840 [4,3,2,2,2,2,1] => 43680 [4,3,2,2,2,1,1,1] => 43680 [4,3,2,2,1,1,1,1,1] => 43680 [4,3,2,1,1,1,1,1,1,1] => 43680 [4,3,1,1,1,1,1,1,1,1,1] => 21840 [4,2,2,2,2,2,2] => 10920 [4,2,2,2,2,2,1,1] => 21840 [4,2,2,2,2,1,1,1,1] => 21840 [4,2,2,2,1,1,1,1,1,1] => 21840 [4,2,2,1,1,1,1,1,1,1,1] => 21840 [4,2,1,1,1,1,1,1,1,1,1,1] => 21840 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 7280 [3,3,3,3,3,1] => 1680 [3,3,3,3,2,2] => 1680 [3,3,3,3,2,1,1] => 3360 [3,3,3,3,1,1,1,1] => 1680 [3,3,3,2,2,2,1] => 3360 [3,3,3,2,2,1,1,1] => 3360 [3,3,3,2,1,1,1,1,1] => 3360 [3,3,3,1,1,1,1,1,1,1] => 1680 [3,3,2,2,2,2,2] => 1680 [3,3,2,2,2,2,1,1] => 3360 [3,3,2,2,2,1,1,1,1] => 3360 [3,3,2,2,1,1,1,1,1,1] => 3360 [3,3,2,1,1,1,1,1,1,1,1] => 3360 [3,3,1,1,1,1,1,1,1,1,1,1] => 1680 [3,2,2,2,2,2,2,1] => 3360 [3,2,2,2,2,2,1,1,1] => 3360 [3,2,2,2,2,1,1,1,1,1] => 3360 [3,2,2,2,1,1,1,1,1,1,1] => 3360 [3,2,2,1,1,1,1,1,1,1,1,1] => 3360 [3,2,1,1,1,1,1,1,1,1,1,1,1] => 3360 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1680 [2,2,2,2,2,2,2,2] => 120 [2,2,2,2,2,2,2,1,1] => 240 [2,2,2,2,2,2,1,1,1,1] => 240 [2,2,2,2,2,1,1,1,1,1,1] => 240 [2,2,2,2,1,1,1,1,1,1,1,1] => 240 [2,2,2,1,1,1,1,1,1,1,1,1,1] => 240 [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 240 [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 240 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16 ----------------------------------------------------------------------------- Created: Apr 30, 2019 at 11:28 by Per Alexandersson ----------------------------------------------------------------------------- Last Updated: Apr 23, 2022 at 10:56 by Martin Rubey