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Identifier
  • St000055: Permutations ⟶ ℤ (values match St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$.)
Values
=>
[1]=>0 [1,2]=>0 [2,1]=>1 [1,2,3]=>0 [1,3,2]=>1 [2,1,3]=>1 [2,3,1]=>3 [3,1,2]=>3 [3,2,1]=>4 [1,2,3,4]=>0 [1,2,4,3]=>1 [1,3,2,4]=>1 [1,3,4,2]=>3 [1,4,2,3]=>3 [1,4,3,2]=>4 [2,1,3,4]=>1 [2,1,4,3]=>2 [2,3,1,4]=>3 [2,3,4,1]=>6 [2,4,1,3]=>5 [2,4,3,1]=>7 [3,1,2,4]=>3 [3,1,4,2]=>5 [3,2,1,4]=>4 [3,2,4,1]=>7 [3,4,1,2]=>8 [3,4,2,1]=>9 [4,1,2,3]=>6 [4,1,3,2]=>7 [4,2,1,3]=>7 [4,2,3,1]=>9 [4,3,1,2]=>9 [4,3,2,1]=>10 [1,2,3,4,5]=>0 [1,2,3,5,4]=>1 [1,2,4,3,5]=>1 [1,2,4,5,3]=>3 [1,2,5,3,4]=>3 [1,2,5,4,3]=>4 [1,3,2,4,5]=>1 [1,3,2,5,4]=>2 [1,3,4,2,5]=>3 [1,3,4,5,2]=>6 [1,3,5,2,4]=>5 [1,3,5,4,2]=>7 [1,4,2,3,5]=>3 [1,4,2,5,3]=>5 [1,4,3,2,5]=>4 [1,4,3,5,2]=>7 [1,4,5,2,3]=>8 [1,4,5,3,2]=>9 [1,5,2,3,4]=>6 [1,5,2,4,3]=>7 [1,5,3,2,4]=>7 [1,5,3,4,2]=>9 [1,5,4,2,3]=>9 [1,5,4,3,2]=>10 [2,1,3,4,5]=>1 [2,1,3,5,4]=>2 [2,1,4,3,5]=>2 [2,1,4,5,3]=>4 [2,1,5,3,4]=>4 [2,1,5,4,3]=>5 [2,3,1,4,5]=>3 [2,3,1,5,4]=>4 [2,3,4,1,5]=>6 [2,3,4,5,1]=>10 [2,3,5,1,4]=>8 [2,3,5,4,1]=>11 [2,4,1,3,5]=>5 [2,4,1,5,3]=>7 [2,4,3,1,5]=>7 [2,4,3,5,1]=>11 [2,4,5,1,3]=>11 [2,4,5,3,1]=>13 [2,5,1,3,4]=>8 [2,5,1,4,3]=>9 [2,5,3,1,4]=>10 [2,5,3,4,1]=>13 [2,5,4,1,3]=>12 [2,5,4,3,1]=>14 [3,1,2,4,5]=>3 [3,1,2,5,4]=>4 [3,1,4,2,5]=>5 [3,1,4,5,2]=>8 [3,1,5,2,4]=>7 [3,1,5,4,2]=>9 [3,2,1,4,5]=>4 [3,2,1,5,4]=>5 [3,2,4,1,5]=>7 [3,2,4,5,1]=>11 [3,2,5,1,4]=>9 [3,2,5,4,1]=>12 [3,4,1,2,5]=>8 [3,4,1,5,2]=>11 [3,4,2,1,5]=>9 [3,4,2,5,1]=>13 [3,4,5,1,2]=>15 [3,4,5,2,1]=>16 [3,5,1,2,4]=>11 [3,5,1,4,2]=>13 [3,5,2,1,4]=>12 [3,5,2,4,1]=>15 [3,5,4,1,2]=>16 [3,5,4,2,1]=>17 [4,1,2,3,5]=>6 [4,1,2,5,3]=>8 [4,1,3,2,5]=>7 [4,1,3,5,2]=>10 [4,1,5,2,3]=>11 [4,1,5,3,2]=>12 [4,2,1,3,5]=>7 [4,2,1,5,3]=>9 [4,2,3,1,5]=>9 [4,2,3,5,1]=>13 [4,2,5,1,3]=>13 [4,2,5,3,1]=>15 [4,3,1,2,5]=>9 [4,3,1,5,2]=>12 [4,3,2,1,5]=>10 [4,3,2,5,1]=>14 [4,3,5,1,2]=>16 [4,3,5,2,1]=>17 [4,5,1,2,3]=>15 [4,5,1,3,2]=>16 [4,5,2,1,3]=>16 [4,5,2,3,1]=>18 [4,5,3,1,2]=>18 [4,5,3,2,1]=>19 [5,1,2,3,4]=>10 [5,1,2,4,3]=>11 [5,1,3,2,4]=>11 [5,1,3,4,2]=>13 [5,1,4,2,3]=>13 [5,1,4,3,2]=>14 [5,2,1,3,4]=>11 [5,2,1,4,3]=>12 [5,2,3,1,4]=>13 [5,2,3,4,1]=>16 [5,2,4,1,3]=>15 [5,2,4,3,1]=>17 [5,3,1,2,4]=>13 [5,3,1,4,2]=>15 [5,3,2,1,4]=>14 [5,3,2,4,1]=>17 [5,3,4,1,2]=>18 [5,3,4,2,1]=>19 [5,4,1,2,3]=>16 [5,4,1,3,2]=>17 [5,4,2,1,3]=>17 [5,4,2,3,1]=>19 [5,4,3,1,2]=>19 [5,4,3,2,1]=>20 [1,2,3,4,5,6]=>0 [1,2,3,4,6,5]=>1 [1,2,3,5,4,6]=>1 [1,2,3,5,6,4]=>3 [1,2,3,6,4,5]=>3 [1,2,3,6,5,4]=>4 [1,2,4,3,5,6]=>1 [1,2,4,3,6,5]=>2 [1,2,4,5,3,6]=>3 [1,2,4,5,6,3]=>6 [1,2,4,6,3,5]=>5 [1,2,4,6,5,3]=>7 [1,2,5,3,4,6]=>3 [1,2,5,3,6,4]=>5 [1,2,5,4,3,6]=>4 [1,2,5,4,6,3]=>7 [1,2,5,6,3,4]=>8 [1,2,5,6,4,3]=>9 [1,2,6,3,4,5]=>6 [1,2,6,3,5,4]=>7 [1,2,6,4,3,5]=>7 [1,2,6,4,5,3]=>9 [1,2,6,5,3,4]=>9 [1,2,6,5,4,3]=>10 [1,3,2,4,5,6]=>1 [1,3,2,4,6,5]=>2 [1,3,2,5,4,6]=>2 [1,3,2,5,6,4]=>4 [1,3,2,6,4,5]=>4 [1,3,2,6,5,4]=>5 [1,3,4,2,5,6]=>3 [1,3,4,2,6,5]=>4 [1,3,4,5,2,6]=>6 [1,3,4,5,6,2]=>10 [1,3,4,6,2,5]=>8 [1,3,4,6,5,2]=>11 [1,3,5,2,4,6]=>5 [1,3,5,2,6,4]=>7 [1,3,5,4,2,6]=>7 [1,3,5,4,6,2]=>11 [1,3,5,6,2,4]=>11 [1,3,5,6,4,2]=>13 [1,3,6,2,4,5]=>8 [1,3,6,2,5,4]=>9 [1,3,6,4,2,5]=>10 [1,3,6,4,5,2]=>13 [1,3,6,5,2,4]=>12 [1,3,6,5,4,2]=>14 [1,4,2,3,5,6]=>3 [1,4,2,3,6,5]=>4 [1,4,2,5,3,6]=>5 [1,4,2,5,6,3]=>8 [1,4,2,6,3,5]=>7 [1,4,2,6,5,3]=>9 [1,4,3,2,5,6]=>4 [1,4,3,2,6,5]=>5 [1,4,3,5,2,6]=>7 [1,4,3,5,6,2]=>11 [1,4,3,6,2,5]=>9 [1,4,3,6,5,2]=>12 [1,4,5,2,3,6]=>8 [1,4,5,2,6,3]=>11 [1,4,5,3,2,6]=>9 [1,4,5,3,6,2]=>13 [1,4,5,6,2,3]=>15 [1,4,5,6,3,2]=>16 [1,4,6,2,3,5]=>11 [1,4,6,2,5,3]=>13 [1,4,6,3,2,5]=>12 [1,4,6,3,5,2]=>15 [1,4,6,5,2,3]=>16 [1,4,6,5,3,2]=>17 [1,5,2,3,4,6]=>6 [1,5,2,3,6,4]=>8 [1,5,2,4,3,6]=>7 [1,5,2,4,6,3]=>10 [1,5,2,6,3,4]=>11 [1,5,2,6,4,3]=>12 [1,5,3,2,4,6]=>7 [1,5,3,2,6,4]=>9 [1,5,3,4,2,6]=>9 [1,5,3,4,6,2]=>13 [1,5,3,6,2,4]=>13 [1,5,3,6,4,2]=>15 [1,5,4,2,3,6]=>9 [1,5,4,2,6,3]=>12 [1,5,4,3,2,6]=>10 [1,5,4,3,6,2]=>14 [1,5,4,6,2,3]=>16 [1,5,4,6,3,2]=>17 [1,5,6,2,3,4]=>15 [1,5,6,2,4,3]=>16 [1,5,6,3,2,4]=>16 [1,5,6,3,4,2]=>18 [1,5,6,4,2,3]=>18 [1,5,6,4,3,2]=>19 [1,6,2,3,4,5]=>10 [1,6,2,3,5,4]=>11 [1,6,2,4,3,5]=>11 [1,6,2,4,5,3]=>13 [1,6,2,5,3,4]=>13 [1,6,2,5,4,3]=>14 [1,6,3,2,4,5]=>11 [1,6,3,2,5,4]=>12 [1,6,3,4,2,5]=>13 [1,6,3,4,5,2]=>16 [1,6,3,5,2,4]=>15 [1,6,3,5,4,2]=>17 [1,6,4,2,3,5]=>13 [1,6,4,2,5,3]=>15 [1,6,4,3,2,5]=>14 [1,6,4,3,5,2]=>17 [1,6,4,5,2,3]=>18 [1,6,4,5,3,2]=>19 [1,6,5,2,3,4]=>16 [1,6,5,2,4,3]=>17 [1,6,5,3,2,4]=>17 [1,6,5,3,4,2]=>19 [1,6,5,4,2,3]=>19 [1,6,5,4,3,2]=>20 [2,1,3,4,5,6]=>1 [2,1,3,4,6,5]=>2 [2,1,3,5,4,6]=>2 [2,1,3,5,6,4]=>4 [2,1,3,6,4,5]=>4 [2,1,3,6,5,4]=>5 [2,1,4,3,5,6]=>2 [2,1,4,3,6,5]=>3 [2,1,4,5,3,6]=>4 [2,1,4,5,6,3]=>7 [2,1,4,6,3,5]=>6 [2,1,4,6,5,3]=>8 [2,1,5,3,4,6]=>4 [2,1,5,3,6,4]=>6 [2,1,5,4,3,6]=>5 [2,1,5,4,6,3]=>8 [2,1,5,6,3,4]=>9 [2,1,5,6,4,3]=>10 [2,1,6,3,4,5]=>7 [2,1,6,3,5,4]=>8 [2,1,6,4,3,5]=>8 [2,1,6,4,5,3]=>10 [2,1,6,5,3,4]=>10 [2,1,6,5,4,3]=>11 [2,3,1,4,5,6]=>3 [2,3,1,4,6,5]=>4 [2,3,1,5,4,6]=>4 [2,3,1,5,6,4]=>6 [2,3,1,6,4,5]=>6 [2,3,1,6,5,4]=>7 [2,3,4,1,5,6]=>6 [2,3,4,1,6,5]=>7 [2,3,4,5,1,6]=>10 [2,3,4,5,6,1]=>15 [2,3,4,6,1,5]=>12 [2,3,4,6,5,1]=>16 [2,3,5,1,4,6]=>8 [2,3,5,1,6,4]=>10 [2,3,5,4,1,6]=>11 [2,3,5,4,6,1]=>16 [2,3,5,6,1,4]=>15 [2,3,5,6,4,1]=>18 [2,3,6,1,4,5]=>11 [2,3,6,1,5,4]=>12 [2,3,6,4,1,5]=>14 [2,3,6,4,5,1]=>18 [2,3,6,5,1,4]=>16 [2,3,6,5,4,1]=>19 [2,4,1,3,5,6]=>5 [2,4,1,3,6,5]=>6 [2,4,1,5,3,6]=>7 [2,4,1,5,6,3]=>10 [2,4,1,6,3,5]=>9 [2,4,1,6,5,3]=>11 [2,4,3,1,5,6]=>7 [2,4,3,1,6,5]=>8 [2,4,3,5,1,6]=>11 [2,4,3,5,6,1]=>16 [2,4,3,6,1,5]=>13 [2,4,3,6,5,1]=>17 [2,4,5,1,3,6]=>11 [2,4,5,1,6,3]=>14 [2,4,5,3,1,6]=>13 [2,4,5,3,6,1]=>18 [2,4,5,6,1,3]=>19 [2,4,5,6,3,1]=>21 [2,4,6,1,3,5]=>14 [2,4,6,1,5,3]=>16 [2,4,6,3,1,5]=>16 [2,4,6,3,5,1]=>20 [2,4,6,5,1,3]=>20 [2,4,6,5,3,1]=>22 [2,5,1,3,4,6]=>8 [2,5,1,3,6,4]=>10 [2,5,1,4,3,6]=>9 [2,5,1,4,6,3]=>12 [2,5,1,6,3,4]=>13 [2,5,1,6,4,3]=>14 [2,5,3,1,4,6]=>10 [2,5,3,1,6,4]=>12 [2,5,3,4,1,6]=>13 [2,5,3,4,6,1]=>18 [2,5,3,6,1,4]=>17 [2,5,3,6,4,1]=>20 [2,5,4,1,3,6]=>12 [2,5,4,1,6,3]=>15 [2,5,4,3,1,6]=>14 [2,5,4,3,6,1]=>19 [2,5,4,6,1,3]=>20 [2,5,4,6,3,1]=>22 [2,5,6,1,3,4]=>18 [2,5,6,1,4,3]=>19 [2,5,6,3,1,4]=>20 [2,5,6,3,4,1]=>23 [2,5,6,4,1,3]=>22 [2,5,6,4,3,1]=>24 [2,6,1,3,4,5]=>12 [2,6,1,3,5,4]=>13 [2,6,1,4,3,5]=>13 [2,6,1,4,5,3]=>15 [2,6,1,5,3,4]=>15 [2,6,1,5,4,3]=>16 [2,6,3,1,4,5]=>14 [2,6,3,1,5,4]=>15 [2,6,3,4,1,5]=>17 [2,6,3,4,5,1]=>21 [2,6,3,5,1,4]=>19 [2,6,3,5,4,1]=>22 [2,6,4,1,3,5]=>16 [2,6,4,1,5,3]=>18 [2,6,4,3,1,5]=>18 [2,6,4,3,5,1]=>22 [2,6,4,5,1,3]=>22 [2,6,4,5,3,1]=>24 [2,6,5,1,3,4]=>19 [2,6,5,1,4,3]=>20 [2,6,5,3,1,4]=>21 [2,6,5,3,4,1]=>24 [2,6,5,4,1,3]=>23 [2,6,5,4,3,1]=>25 [3,1,2,4,5,6]=>3 [3,1,2,4,6,5]=>4 [3,1,2,5,4,6]=>4 [3,1,2,5,6,4]=>6 [3,1,2,6,4,5]=>6 [3,1,2,6,5,4]=>7 [3,1,4,2,5,6]=>5 [3,1,4,2,6,5]=>6 [3,1,4,5,2,6]=>8 [3,1,4,5,6,2]=>12 [3,1,4,6,2,5]=>10 [3,1,4,6,5,2]=>13 [3,1,5,2,4,6]=>7 [3,1,5,2,6,4]=>9 [3,1,5,4,2,6]=>9 [3,1,5,4,6,2]=>13 [3,1,5,6,2,4]=>13 [3,1,5,6,4,2]=>15 [3,1,6,2,4,5]=>10 [3,1,6,2,5,4]=>11 [3,1,6,4,2,5]=>12 [3,1,6,4,5,2]=>15 [3,1,6,5,2,4]=>14 [3,1,6,5,4,2]=>16 [3,2,1,4,5,6]=>4 [3,2,1,4,6,5]=>5 [3,2,1,5,4,6]=>5 [3,2,1,5,6,4]=>7 [3,2,1,6,4,5]=>7 [3,2,1,6,5,4]=>8 [3,2,4,1,5,6]=>7 [3,2,4,1,6,5]=>8 [3,2,4,5,1,6]=>11 [3,2,4,5,6,1]=>16 [3,2,4,6,1,5]=>13 [3,2,4,6,5,1]=>17 [3,2,5,1,4,6]=>9 [3,2,5,1,6,4]=>11 [3,2,5,4,1,6]=>12 [3,2,5,4,6,1]=>17 [3,2,5,6,1,4]=>16 [3,2,5,6,4,1]=>19 [3,2,6,1,4,5]=>12 [3,2,6,1,5,4]=>13 [3,2,6,4,1,5]=>15 [3,2,6,4,5,1]=>19 [3,2,6,5,1,4]=>17 [3,2,6,5,4,1]=>20 [3,4,1,2,5,6]=>8 [3,4,1,2,6,5]=>9 [3,4,1,5,2,6]=>11 [3,4,1,5,6,2]=>15 [3,4,1,6,2,5]=>13 [3,4,1,6,5,2]=>16 [3,4,2,1,5,6]=>9 [3,4,2,1,6,5]=>10 [3,4,2,5,1,6]=>13 [3,4,2,5,6,1]=>18 [3,4,2,6,1,5]=>15 [3,4,2,6,5,1]=>19 [3,4,5,1,2,6]=>15 [3,4,5,1,6,2]=>19 [3,4,5,2,1,6]=>16 [3,4,5,2,6,1]=>21 [3,4,5,6,1,2]=>24 [3,4,5,6,2,1]=>25 [3,4,6,1,2,5]=>18 [3,4,6,1,5,2]=>21 [3,4,6,2,1,5]=>19 [3,4,6,2,5,1]=>23 [3,4,6,5,1,2]=>25 [3,4,6,5,2,1]=>26 [3,5,1,2,4,6]=>11 [3,5,1,2,6,4]=>13 [3,5,1,4,2,6]=>13 [3,5,1,4,6,2]=>17 [3,5,1,6,2,4]=>17 [3,5,1,6,4,2]=>19 [3,5,2,1,4,6]=>12 [3,5,2,1,6,4]=>14 [3,5,2,4,1,6]=>15 [3,5,2,4,6,1]=>20 [3,5,2,6,1,4]=>19 [3,5,2,6,4,1]=>22 [3,5,4,1,2,6]=>16 [3,5,4,1,6,2]=>20 [3,5,4,2,1,6]=>17 [3,5,4,2,6,1]=>22 [3,5,4,6,1,2]=>25 [3,5,4,6,2,1]=>26 [3,5,6,1,2,4]=>22 [3,5,6,1,4,2]=>24 [3,5,6,2,1,4]=>23 [3,5,6,2,4,1]=>26 [3,5,6,4,1,2]=>27 [3,5,6,4,2,1]=>28 [3,6,1,2,4,5]=>15 [3,6,1,2,5,4]=>16 [3,6,1,4,2,5]=>17 [3,6,1,4,5,2]=>20 [3,6,1,5,2,4]=>19 [3,6,1,5,4,2]=>21 [3,6,2,1,4,5]=>16 [3,6,2,1,5,4]=>17 [3,6,2,4,1,5]=>19 [3,6,2,4,5,1]=>23 [3,6,2,5,1,4]=>21 [3,6,2,5,4,1]=>24 [3,6,4,1,2,5]=>20 [3,6,4,1,5,2]=>23 [3,6,4,2,1,5]=>21 [3,6,4,2,5,1]=>25 [3,6,4,5,1,2]=>27 [3,6,4,5,2,1]=>28 [3,6,5,1,2,4]=>23 [3,6,5,1,4,2]=>25 [3,6,5,2,1,4]=>24 [3,6,5,2,4,1]=>27 [3,6,5,4,1,2]=>28 [3,6,5,4,2,1]=>29 [4,1,2,3,5,6]=>6 [4,1,2,3,6,5]=>7 [4,1,2,5,3,6]=>8 [4,1,2,5,6,3]=>11 [4,1,2,6,3,5]=>10 [4,1,2,6,5,3]=>12 [4,1,3,2,5,6]=>7 [4,1,3,2,6,5]=>8 [4,1,3,5,2,6]=>10 [4,1,3,5,6,2]=>14 [4,1,3,6,2,5]=>12 [4,1,3,6,5,2]=>15 [4,1,5,2,3,6]=>11 [4,1,5,2,6,3]=>14 [4,1,5,3,2,6]=>12 [4,1,5,3,6,2]=>16 [4,1,5,6,2,3]=>18 [4,1,5,6,3,2]=>19 [4,1,6,2,3,5]=>14 [4,1,6,2,5,3]=>16 [4,1,6,3,2,5]=>15 [4,1,6,3,5,2]=>18 [4,1,6,5,2,3]=>19 [4,1,6,5,3,2]=>20 [4,2,1,3,5,6]=>7 [4,2,1,3,6,5]=>8 [4,2,1,5,3,6]=>9 [4,2,1,5,6,3]=>12 [4,2,1,6,3,5]=>11 [4,2,1,6,5,3]=>13 [4,2,3,1,5,6]=>9 [4,2,3,1,6,5]=>10 [4,2,3,5,1,6]=>13 [4,2,3,5,6,1]=>18 [4,2,3,6,1,5]=>15 [4,2,3,6,5,1]=>19 [4,2,5,1,3,6]=>13 [4,2,5,1,6,3]=>16 [4,2,5,3,1,6]=>15 [4,2,5,3,6,1]=>20 [4,2,5,6,1,3]=>21 [4,2,5,6,3,1]=>23 [4,2,6,1,3,5]=>16 [4,2,6,1,5,3]=>18 [4,2,6,3,1,5]=>18 [4,2,6,3,5,1]=>22 [4,2,6,5,1,3]=>22 [4,2,6,5,3,1]=>24 [4,3,1,2,5,6]=>9 [4,3,1,2,6,5]=>10 [4,3,1,5,2,6]=>12 [4,3,1,5,6,2]=>16 [4,3,1,6,2,5]=>14 [4,3,1,6,5,2]=>17 [4,3,2,1,5,6]=>10 [4,3,2,1,6,5]=>11 [4,3,2,5,1,6]=>14 [4,3,2,5,6,1]=>19 [4,3,2,6,1,5]=>16 [4,3,2,6,5,1]=>20 [4,3,5,1,2,6]=>16 [4,3,5,1,6,2]=>20 [4,3,5,2,1,6]=>17 [4,3,5,2,6,1]=>22 [4,3,5,6,1,2]=>25 [4,3,5,6,2,1]=>26 [4,3,6,1,2,5]=>19 [4,3,6,1,5,2]=>22 [4,3,6,2,1,5]=>20 [4,3,6,2,5,1]=>24 [4,3,6,5,1,2]=>26 [4,3,6,5,2,1]=>27 [4,5,1,2,3,6]=>15 [4,5,1,2,6,3]=>18 [4,5,1,3,2,6]=>16 [4,5,1,3,6,2]=>20 [4,5,1,6,2,3]=>22 [4,5,1,6,3,2]=>23 [4,5,2,1,3,6]=>16 [4,5,2,1,6,3]=>19 [4,5,2,3,1,6]=>18 [4,5,2,3,6,1]=>23 [4,5,2,6,1,3]=>24 [4,5,2,6,3,1]=>26 [4,5,3,1,2,6]=>18 [4,5,3,1,6,2]=>22 [4,5,3,2,1,6]=>19 [4,5,3,2,6,1]=>24 [4,5,3,6,1,2]=>27 [4,5,3,6,2,1]=>28 [4,5,6,1,2,3]=>27 [4,5,6,1,3,2]=>28 [4,5,6,2,1,3]=>28 [4,5,6,2,3,1]=>30 [4,5,6,3,1,2]=>30 [4,5,6,3,2,1]=>31 [4,6,1,2,3,5]=>19 [4,6,1,2,5,3]=>21 [4,6,1,3,2,5]=>20 [4,6,1,3,5,2]=>23 [4,6,1,5,2,3]=>24 [4,6,1,5,3,2]=>25 [4,6,2,1,3,5]=>20 [4,6,2,1,5,3]=>22 [4,6,2,3,1,5]=>22 [4,6,2,3,5,1]=>26 [4,6,2,5,1,3]=>26 [4,6,2,5,3,1]=>28 [4,6,3,1,2,5]=>22 [4,6,3,1,5,2]=>25 [4,6,3,2,1,5]=>23 [4,6,3,2,5,1]=>27 [4,6,3,5,1,2]=>29 [4,6,3,5,2,1]=>30 [4,6,5,1,2,3]=>28 [4,6,5,1,3,2]=>29 [4,6,5,2,1,3]=>29 [4,6,5,2,3,1]=>31 [4,6,5,3,1,2]=>31 [4,6,5,3,2,1]=>32 [5,1,2,3,4,6]=>10 [5,1,2,3,6,4]=>12 [5,1,2,4,3,6]=>11 [5,1,2,4,6,3]=>14 [5,1,2,6,3,4]=>15 [5,1,2,6,4,3]=>16 [5,1,3,2,4,6]=>11 [5,1,3,2,6,4]=>13 [5,1,3,4,2,6]=>13 [5,1,3,4,6,2]=>17 [5,1,3,6,2,4]=>17 [5,1,3,6,4,2]=>19 [5,1,4,2,3,6]=>13 [5,1,4,2,6,3]=>16 [5,1,4,3,2,6]=>14 [5,1,4,3,6,2]=>18 [5,1,4,6,2,3]=>20 [5,1,4,6,3,2]=>21 [5,1,6,2,3,4]=>19 [5,1,6,2,4,3]=>20 [5,1,6,3,2,4]=>20 [5,1,6,3,4,2]=>22 [5,1,6,4,2,3]=>22 [5,1,6,4,3,2]=>23 [5,2,1,3,4,6]=>11 [5,2,1,3,6,4]=>13 [5,2,1,4,3,6]=>12 [5,2,1,4,6,3]=>15 [5,2,1,6,3,4]=>16 [5,2,1,6,4,3]=>17 [5,2,3,1,4,6]=>13 [5,2,3,1,6,4]=>15 [5,2,3,4,1,6]=>16 [5,2,3,4,6,1]=>21 [5,2,3,6,1,4]=>20 [5,2,3,6,4,1]=>23 [5,2,4,1,3,6]=>15 [5,2,4,1,6,3]=>18 [5,2,4,3,1,6]=>17 [5,2,4,3,6,1]=>22 [5,2,4,6,1,3]=>23 [5,2,4,6,3,1]=>25 [5,2,6,1,3,4]=>21 [5,2,6,1,4,3]=>22 [5,2,6,3,1,4]=>23 [5,2,6,3,4,1]=>26 [5,2,6,4,1,3]=>25 [5,2,6,4,3,1]=>27 [5,3,1,2,4,6]=>13 [5,3,1,2,6,4]=>15 [5,3,1,4,2,6]=>15 [5,3,1,4,6,2]=>19 [5,3,1,6,2,4]=>19 [5,3,1,6,4,2]=>21 [5,3,2,1,4,6]=>14 [5,3,2,1,6,4]=>16 [5,3,2,4,1,6]=>17 [5,3,2,4,6,1]=>22 [5,3,2,6,1,4]=>21 [5,3,2,6,4,1]=>24 [5,3,4,1,2,6]=>18 [5,3,4,1,6,2]=>22 [5,3,4,2,1,6]=>19 [5,3,4,2,6,1]=>24 [5,3,4,6,1,2]=>27 [5,3,4,6,2,1]=>28 [5,3,6,1,2,4]=>24 [5,3,6,1,4,2]=>26 [5,3,6,2,1,4]=>25 [5,3,6,2,4,1]=>28 [5,3,6,4,1,2]=>29 [5,3,6,4,2,1]=>30 [5,4,1,2,3,6]=>16 [5,4,1,2,6,3]=>19 [5,4,1,3,2,6]=>17 [5,4,1,3,6,2]=>21 [5,4,1,6,2,3]=>23 [5,4,1,6,3,2]=>24 [5,4,2,1,3,6]=>17 [5,4,2,1,6,3]=>20 [5,4,2,3,1,6]=>19 [5,4,2,3,6,1]=>24 [5,4,2,6,1,3]=>25 [5,4,2,6,3,1]=>27 [5,4,3,1,2,6]=>19 [5,4,3,1,6,2]=>23 [5,4,3,2,1,6]=>20 [5,4,3,2,6,1]=>25 [5,4,3,6,1,2]=>28 [5,4,3,6,2,1]=>29 [5,4,6,1,2,3]=>28 [5,4,6,1,3,2]=>29 [5,4,6,2,1,3]=>29 [5,4,6,2,3,1]=>31 [5,4,6,3,1,2]=>31 [5,4,6,3,2,1]=>32 [5,6,1,2,3,4]=>24 [5,6,1,2,4,3]=>25 [5,6,1,3,2,4]=>25 [5,6,1,3,4,2]=>27 [5,6,1,4,2,3]=>27 [5,6,1,4,3,2]=>28 [5,6,2,1,3,4]=>25 [5,6,2,1,4,3]=>26 [5,6,2,3,1,4]=>27 [5,6,2,3,4,1]=>30 [5,6,2,4,1,3]=>29 [5,6,2,4,3,1]=>31 [5,6,3,1,2,4]=>27 [5,6,3,1,4,2]=>29 [5,6,3,2,1,4]=>28 [5,6,3,2,4,1]=>31 [5,6,3,4,1,2]=>32 [5,6,3,4,2,1]=>33 [5,6,4,1,2,3]=>30 [5,6,4,1,3,2]=>31 [5,6,4,2,1,3]=>31 [5,6,4,2,3,1]=>33 [5,6,4,3,1,2]=>33 [5,6,4,3,2,1]=>34 [6,1,2,3,4,5]=>15 [6,1,2,3,5,4]=>16 [6,1,2,4,3,5]=>16 [6,1,2,4,5,3]=>18 [6,1,2,5,3,4]=>18 [6,1,2,5,4,3]=>19 [6,1,3,2,4,5]=>16 [6,1,3,2,5,4]=>17 [6,1,3,4,2,5]=>18 [6,1,3,4,5,2]=>21 [6,1,3,5,2,4]=>20 [6,1,3,5,4,2]=>22 [6,1,4,2,3,5]=>18 [6,1,4,2,5,3]=>20 [6,1,4,3,2,5]=>19 [6,1,4,3,5,2]=>22 [6,1,4,5,2,3]=>23 [6,1,4,5,3,2]=>24 [6,1,5,2,3,4]=>21 [6,1,5,2,4,3]=>22 [6,1,5,3,2,4]=>22 [6,1,5,3,4,2]=>24 [6,1,5,4,2,3]=>24 [6,1,5,4,3,2]=>25 [6,2,1,3,4,5]=>16 [6,2,1,3,5,4]=>17 [6,2,1,4,3,5]=>17 [6,2,1,4,5,3]=>19 [6,2,1,5,3,4]=>19 [6,2,1,5,4,3]=>20 [6,2,3,1,4,5]=>18 [6,2,3,1,5,4]=>19 [6,2,3,4,1,5]=>21 [6,2,3,4,5,1]=>25 [6,2,3,5,1,4]=>23 [6,2,3,5,4,1]=>26 [6,2,4,1,3,5]=>20 [6,2,4,1,5,3]=>22 [6,2,4,3,1,5]=>22 [6,2,4,3,5,1]=>26 [6,2,4,5,1,3]=>26 [6,2,4,5,3,1]=>28 [6,2,5,1,3,4]=>23 [6,2,5,1,4,3]=>24 [6,2,5,3,1,4]=>25 [6,2,5,3,4,1]=>28 [6,2,5,4,1,3]=>27 [6,2,5,4,3,1]=>29 [6,3,1,2,4,5]=>18 [6,3,1,2,5,4]=>19 [6,3,1,4,2,5]=>20 [6,3,1,4,5,2]=>23 [6,3,1,5,2,4]=>22 [6,3,1,5,4,2]=>24 [6,3,2,1,4,5]=>19 [6,3,2,1,5,4]=>20 [6,3,2,4,1,5]=>22 [6,3,2,4,5,1]=>26 [6,3,2,5,1,4]=>24 [6,3,2,5,4,1]=>27 [6,3,4,1,2,5]=>23 [6,3,4,1,5,2]=>26 [6,3,4,2,1,5]=>24 [6,3,4,2,5,1]=>28 [6,3,4,5,1,2]=>30 [6,3,4,5,2,1]=>31 [6,3,5,1,2,4]=>26 [6,3,5,1,4,2]=>28 [6,3,5,2,1,4]=>27 [6,3,5,2,4,1]=>30 [6,3,5,4,1,2]=>31 [6,3,5,4,2,1]=>32 [6,4,1,2,3,5]=>21 [6,4,1,2,5,3]=>23 [6,4,1,3,2,5]=>22 [6,4,1,3,5,2]=>25 [6,4,1,5,2,3]=>26 [6,4,1,5,3,2]=>27 [6,4,2,1,3,5]=>22 [6,4,2,1,5,3]=>24 [6,4,2,3,1,5]=>24 [6,4,2,3,5,1]=>28 [6,4,2,5,1,3]=>28 [6,4,2,5,3,1]=>30 [6,4,3,1,2,5]=>24 [6,4,3,1,5,2]=>27 [6,4,3,2,1,5]=>25 [6,4,3,2,5,1]=>29 [6,4,3,5,1,2]=>31 [6,4,3,5,2,1]=>32 [6,4,5,1,2,3]=>30 [6,4,5,1,3,2]=>31 [6,4,5,2,1,3]=>31 [6,4,5,2,3,1]=>33 [6,4,5,3,1,2]=>33 [6,4,5,3,2,1]=>34 [6,5,1,2,3,4]=>25 [6,5,1,2,4,3]=>26 [6,5,1,3,2,4]=>26 [6,5,1,3,4,2]=>28 [6,5,1,4,2,3]=>28 [6,5,1,4,3,2]=>29 [6,5,2,1,3,4]=>26 [6,5,2,1,4,3]=>27 [6,5,2,3,1,4]=>28 [6,5,2,3,4,1]=>31 [6,5,2,4,1,3]=>30 [6,5,2,4,3,1]=>32 [6,5,3,1,2,4]=>28 [6,5,3,1,4,2]=>30 [6,5,3,2,1,4]=>29 [6,5,3,2,4,1]=>32 [6,5,3,4,1,2]=>33 [6,5,3,4,2,1]=>34 [6,5,4,1,2,3]=>31 [6,5,4,1,3,2]=>32 [6,5,4,2,1,3]=>32 [6,5,4,2,3,1]=>34 [6,5,4,3,1,2]=>34 [6,5,4,3,2,1]=>35
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Description
The inversion sum of a permutation.
A pair $a < b$ is an inversion of a permutation $\pi$ if $\pi(a) > \pi(b)$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
This is also half of the metric associated with Spearmans coefficient of association $\rho$, $\sum_i (\pi_i - i)^2$, see [5].
This is also equal to the total number of occurrences of the classical permutation patterns $[2,1], [2, 3, 1], [3, 1, 2]$, and $[3, 2, 1]$, see [2].
This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3].
This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding monotone triangle and subtracting $\binom{n+2}{3}$, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of $n$.
This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation $\pi$ in Bruhat order, see Theorem 1 of [6].
References
[1] Lascoux, A., Sch├╝tzenberger, M.-P. Treillis et bases des groupes de Coxeter MathSciNet:1395667
[2] Sack, J., Úlfarsson, H. Refined inversion statistics on permutations MathSciNet:2880660 arXiv:1106.1995
[3] Striker, J. A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux MathSciNet:2794039
[4] a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers. OEIS:A135298
[5] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray MathSciNet:0652736
[6] Kobayashi, M. Enumeration of bigrassmannian permutations below a permutation in Bruhat order arXiv:1005.3335
Code
def statistic(pi):
    return sum( inv[1]-inv[0] for inv in pi.inversions() )

def statistic_alternative(perm):
    pmatrix = perm.to_matrix()
    w = AlternatingSignMatrix(pmatrix).to_monotone_triangle()
    counter = -binomial(len(perm)+2,3)
    for k in [0..(len(w)-1)]:
        for j in [0..(len(w[k])-1)]:
            counter=counter+w[k][j]
    return counter
    
Created
Mar 25, 2013 at 20:49 by Jessica Striker
Updated
May 30, 2019 at 11:28 by Masato Kobayashi