- FindStat

Identifier

Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤ

Values

=>
                            Cc0005;cc-rep-0
                            
                            Cc0007;cc-rep-1
                            
                            Cc0007;cc-rep
                            
                            
                            

                        [1,0]=>[[1],[2]]=>[[1,2]]=>0
[1,0,1,0]=>[[1,3],[2,4]]=>[[1,2,4],[3]]=>2
[1,1,0,0]=>[[1,2],[3,4]]=>[[1,2,3,4]]=>0
[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2,4,6],[3,5]]=>6
[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2,4,5,6],[3]]=>2
[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,2,3,4,6],[5]]=>4
[1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[1,2,3,5,6],[4]]=>3
[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,2,3,4,5,6]]=>0
[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2,4,6,8],[3,5,7]]=>12
[1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[1,2,4,6,7,8],[3,5]]=>6
[1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[1,2,4,5,6,8],[3,7]]=>8
[1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[[1,2,4,5,7,8],[3,6]]=>7
[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[1,2,4,5,6,7,8],[3]]=>2
[1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[1,2,3,4,6,8],[5,7]]=>10
[1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[1,2,3,4,7,8],[5,6]]=>4
[1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[[1,2,3,5,6,8],[4,7]]=>9
[1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[[1,2,3,5,7,8],[4,6]]=>8
[1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[[1,2,3,5,6,7,8],[4]]=>3
[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[1,2,3,4,5,6,8],[7]]=>6
[1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[[1,2,3,4,5,7,8],[6]]=>5
[1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[[1,2,3,4,6,7,8],[5]]=>4
[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[1,2,3,4,5,6,7,8]]=>0
[1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[[1,2,4,6,8,10],[3,5,7,9]]=>20
[1,0,1,0,1,0,1,1,0,0]=>[[1,3,5,7,8],[2,4,6,9,10]]=>[[1,2,4,6,8,9,10],[3,5,7]]=>12
[1,0,1,0,1,1,0,0,1,0]=>[[1,3,5,6,9],[2,4,7,8,10]]=>[[1,2,4,6,7,8,10],[3,5,9]]=>14
[1,0,1,0,1,1,0,1,0,0]=>[[1,3,5,6,8],[2,4,7,9,10]]=>[[1,2,4,6,7,9,10],[3,5,8]]=>13
[1,0,1,0,1,1,1,0,0,0]=>[[1,3,5,6,7],[2,4,8,9,10]]=>[[1,2,4,6,7,8,9,10],[3,5]]=>6
[1,0,1,1,0,0,1,0,1,0]=>[[1,3,4,7,9],[2,5,6,8,10]]=>[[1,2,4,5,6,8,10],[3,7,9]]=>16
[1,0,1,1,0,0,1,1,0,0]=>[[1,3,4,7,8],[2,5,6,9,10]]=>[[1,2,4,5,6,9,10],[3,7,8]]=>8
[1,0,1,1,0,1,0,0,1,0]=>[[1,3,4,6,9],[2,5,7,8,10]]=>[[1,2,4,5,7,8,10],[3,6,9]]=>15
[1,0,1,1,0,1,0,1,0,0]=>[[1,3,4,6,8],[2,5,7,9,10]]=>[[1,2,4,5,7,9,10],[3,6,8]]=>14
[1,0,1,1,0,1,1,0,0,0]=>[[1,3,4,6,7],[2,5,8,9,10]]=>[[1,2,4,5,7,8,9,10],[3,6]]=>7
[1,0,1,1,1,0,0,0,1,0]=>[[1,3,4,5,9],[2,6,7,8,10]]=>[[1,2,4,5,6,7,8,10],[3,9]]=>10
[1,0,1,1,1,0,0,1,0,0]=>[[1,3,4,5,8],[2,6,7,9,10]]=>[[1,2,4,5,6,7,9,10],[3,8]]=>9
[1,0,1,1,1,0,1,0,0,0]=>[[1,3,4,5,7],[2,6,8,9,10]]=>[[1,2,4,5,6,8,9,10],[3,7]]=>8
[1,0,1,1,1,1,0,0,0,0]=>[[1,3,4,5,6],[2,7,8,9,10]]=>[[1,2,4,5,6,7,8,9,10],[3]]=>2
[1,1,0,0,1,0,1,0,1,0]=>[[1,2,5,7,9],[3,4,6,8,10]]=>[[1,2,3,4,6,8,10],[5,7,9]]=>18
[1,1,0,0,1,0,1,1,0,0]=>[[1,2,5,7,8],[3,4,6,9,10]]=>[[1,2,3,4,6,9,10],[5,7,8]]=>10
[1,1,0,0,1,1,0,0,1,0]=>[[1,2,5,6,9],[3,4,7,8,10]]=>[[1,2,3,4,7,8,10],[5,6,9]]=>12
[1,1,0,0,1,1,0,1,0,0]=>[[1,2,5,6,8],[3,4,7,9,10]]=>[[1,2,3,4,7,9,10],[5,6,8]]=>11
[1,1,0,0,1,1,1,0,0,0]=>[[1,2,5,6,7],[3,4,8,9,10]]=>[[1,2,3,4,7,8,9,10],[5,6]]=>4
[1,1,0,1,0,0,1,0,1,0]=>[[1,2,4,7,9],[3,5,6,8,10]]=>[[1,2,3,5,6,8,10],[4,7,9]]=>17
[1,1,0,1,0,0,1,1,0,0]=>[[1,2,4,7,8],[3,5,6,9,10]]=>[[1,2,3,5,6,9,10],[4,7,8]]=>9
[1,1,0,1,0,1,0,0,1,0]=>[[1,2,4,6,9],[3,5,7,8,10]]=>[[1,2,3,5,7,8,10],[4,6,9]]=>16
[1,1,0,1,0,1,0,1,0,0]=>[[1,2,4,6,8],[3,5,7,9,10]]=>[[1,2,3,5,7,9,10],[4,6,8]]=>15
[1,1,0,1,0,1,1,0,0,0]=>[[1,2,4,6,7],[3,5,8,9,10]]=>[[1,2,3,5,7,8,9,10],[4,6]]=>8
[1,1,0,1,1,0,0,0,1,0]=>[[1,2,4,5,9],[3,6,7,8,10]]=>[[1,2,3,5,6,7,8,10],[4,9]]=>11
[1,1,0,1,1,0,0,1,0,0]=>[[1,2,4,5,8],[3,6,7,9,10]]=>[[1,2,3,5,6,7,9,10],[4,8]]=>10
[1,1,0,1,1,0,1,0,0,0]=>[[1,2,4,5,7],[3,6,8,9,10]]=>[[1,2,3,5,6,8,9,10],[4,7]]=>9
[1,1,0,1,1,1,0,0,0,0]=>[[1,2,4,5,6],[3,7,8,9,10]]=>[[1,2,3,5,6,7,8,9,10],[4]]=>3
[1,1,1,0,0,0,1,0,1,0]=>[[1,2,3,7,9],[4,5,6,8,10]]=>[[1,2,3,4,5,6,8,10],[7,9]]=>14
[1,1,1,0,0,0,1,1,0,0]=>[[1,2,3,7,8],[4,5,6,9,10]]=>[[1,2,3,4,5,6,9,10],[7,8]]=>6
[1,1,1,0,0,1,0,0,1,0]=>[[1,2,3,6,9],[4,5,7,8,10]]=>[[1,2,3,4,5,7,8,10],[6,9]]=>13
[1,1,1,0,0,1,0,1,0,0]=>[[1,2,3,6,8],[4,5,7,9,10]]=>[[1,2,3,4,5,7,9,10],[6,8]]=>12
[1,1,1,0,0,1,1,0,0,0]=>[[1,2,3,6,7],[4,5,8,9,10]]=>[[1,2,3,4,5,8,9,10],[6,7]]=>5
[1,1,1,0,1,0,0,0,1,0]=>[[1,2,3,5,9],[4,6,7,8,10]]=>[[1,2,3,4,6,7,8,10],[5,9]]=>12
[1,1,1,0,1,0,0,1,0,0]=>[[1,2,3,5,8],[4,6,7,9,10]]=>[[1,2,3,4,6,7,9,10],[5,8]]=>11
[1,1,1,0,1,0,1,0,0,0]=>[[1,2,3,5,7],[4,6,8,9,10]]=>[[1,2,3,4,6,8,9,10],[5,7]]=>10
[1,1,1,0,1,1,0,0,0,0]=>[[1,2,3,5,6],[4,7,8,9,10]]=>[[1,2,3,4,6,7,8,9,10],[5]]=>4
[1,1,1,1,0,0,0,0,1,0]=>[[1,2,3,4,9],[5,6,7,8,10]]=>[[1,2,3,4,5,6,7,8,10],[9]]=>8
[1,1,1,1,0,0,0,1,0,0]=>[[1,2,3,4,8],[5,6,7,9,10]]=>[[1,2,3,4,5,6,7,9,10],[8]]=>7
[1,1,1,1,0,0,1,0,0,0]=>[[1,2,3,4,7],[5,6,8,9,10]]=>[[1,2,3,4,5,6,8,9,10],[7]]=>6
[1,1,1,1,0,1,0,0,0,0]=>[[1,2,3,4,6],[5,7,8,9,10]]=>[[1,2,3,4,5,7,8,9,10],[6]]=>5
[1,1,1,1,1,0,0,0,0,0]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[[1,2,3,4,5,6,7,8,9,10]]=>0
[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[[1,2,4,6,8,10,12],[3,5,7,9,11]]=>30
[1,0,1,0,1,0,1,0,1,1,0,0]=>[[1,3,5,7,9,10],[2,4,6,8,11,12]]=>[[1,2,4,6,8,10,11,12],[3,5,7,9]]=>20
[1,0,1,0,1,0,1,1,0,0,1,0]=>[[1,3,5,7,8,11],[2,4,6,9,10,12]]=>[[1,2,4,6,8,9,10,12],[3,5,7,11]]=>22
[1,0,1,0,1,0,1,1,0,1,0,0]=>[[1,3,5,7,8,10],[2,4,6,9,11,12]]=>[[1,2,4,6,8,9,11,12],[3,5,7,10]]=>21
[1,0,1,0,1,0,1,1,1,0,0,0]=>[[1,3,5,7,8,9],[2,4,6,10,11,12]]=>[[1,2,4,6,8,9,10,11,12],[3,5,7]]=>12
[1,0,1,0,1,1,0,0,1,0,1,0]=>[[1,3,5,6,9,11],[2,4,7,8,10,12]]=>[[1,2,4,6,7,8,10,12],[3,5,9,11]]=>24
[1,0,1,0,1,1,0,0,1,1,0,0]=>[[1,3,5,6,9,10],[2,4,7,8,11,12]]=>[[1,2,4,6,7,8,11,12],[3,5,9,10]]=>14
[1,0,1,0,1,1,0,1,0,0,1,0]=>[[1,3,5,6,8,11],[2,4,7,9,10,12]]=>[[1,2,4,6,7,9,10,12],[3,5,8,11]]=>23
[1,0,1,0,1,1,0,1,0,1,0,0]=>[[1,3,5,6,8,10],[2,4,7,9,11,12]]=>[[1,2,4,6,7,9,11,12],[3,5,8,10]]=>22
[1,0,1,0,1,1,0,1,1,0,0,0]=>[[1,3,5,6,8,9],[2,4,7,10,11,12]]=>[[1,2,4,6,7,9,10,11,12],[3,5,8]]=>13
[1,0,1,0,1,1,1,0,0,0,1,0]=>[[1,3,5,6,7,11],[2,4,8,9,10,12]]=>[[1,2,4,6,7,8,9,10,12],[3,5,11]]=>16
[1,0,1,0,1,1,1,0,0,1,0,0]=>[[1,3,5,6,7,10],[2,4,8,9,11,12]]=>[[1,2,4,6,7,8,9,11,12],[3,5,10]]=>15
[1,0,1,0,1,1,1,0,1,0,0,0]=>[[1,3,5,6,7,9],[2,4,8,10,11,12]]=>[[1,2,4,6,7,8,10,11,12],[3,5,9]]=>14
[1,0,1,0,1,1,1,1,0,0,0,0]=>[[1,3,5,6,7,8],[2,4,9,10,11,12]]=>[[1,2,4,6,7,8,9,10,11,12],[3,5]]=>6
[1,0,1,1,0,0,1,0,1,0,1,0]=>[[1,3,4,7,9,11],[2,5,6,8,10,12]]=>[[1,2,4,5,6,8,10,12],[3,7,9,11]]=>26
[1,0,1,1,0,0,1,0,1,1,0,0]=>[[1,3,4,7,9,10],[2,5,6,8,11,12]]=>[[1,2,4,5,6,8,11,12],[3,7,9,10]]=>16
[1,0,1,1,0,0,1,1,0,0,1,0]=>[[1,3,4,7,8,11],[2,5,6,9,10,12]]=>[[1,2,4,5,6,9,10,12],[3,7,8,11]]=>18
[1,0,1,1,0,0,1,1,0,1,0,0]=>[[1,3,4,7,8,10],[2,5,6,9,11,12]]=>[[1,2,4,5,6,9,11,12],[3,7,8,10]]=>17
[1,0,1,1,0,0,1,1,1,0,0,0]=>[[1,3,4,7,8,9],[2,5,6,10,11,12]]=>[[1,2,4,5,6,9,10,11,12],[3,7,8]]=>8
[1,0,1,1,0,1,0,0,1,0,1,0]=>[[1,3,4,6,9,11],[2,5,7,8,10,12]]=>[[1,2,4,5,7,8,10,12],[3,6,9,11]]=>25
[1,0,1,1,0,1,0,0,1,1,0,0]=>[[1,3,4,6,9,10],[2,5,7,8,11,12]]=>[[1,2,4,5,7,8,11,12],[3,6,9,10]]=>15
[1,0,1,1,0,1,0,1,0,0,1,0]=>[[1,3,4,6,8,11],[2,5,7,9,10,12]]=>[[1,2,4,5,7,9,10,12],[3,6,8,11]]=>24
[1,0,1,1,0,1,0,1,0,1,0,0]=>[[1,3,4,6,8,10],[2,5,7,9,11,12]]=>[[1,2,4,5,7,9,11,12],[3,6,8,10]]=>23
[1,0,1,1,0,1,0,1,1,0,0,0]=>[[1,3,4,6,8,9],[2,5,7,10,11,12]]=>[[1,2,4,5,7,9,10,11,12],[3,6,8]]=>14
[1,0,1,1,0,1,1,0,0,0,1,0]=>[[1,3,4,6,7,11],[2,5,8,9,10,12]]=>[[1,2,4,5,7,8,9,10,12],[3,6,11]]=>17
[1,0,1,1,0,1,1,0,0,1,0,0]=>[[1,3,4,6,7,10],[2,5,8,9,11,12]]=>[[1,2,4,5,7,8,9,11,12],[3,6,10]]=>16
[1,0,1,1,0,1,1,0,1,0,0,0]=>[[1,3,4,6,7,9],[2,5,8,10,11,12]]=>[[1,2,4,5,7,8,10,11,12],[3,6,9]]=>15
[1,0,1,1,0,1,1,1,0,0,0,0]=>[[1,3,4,6,7,8],[2,5,9,10,11,12]]=>[[1,2,4,5,7,8,9,10,11,12],[3,6]]=>7
[1,0,1,1,1,0,0,0,1,0,1,0]=>[[1,3,4,5,9,11],[2,6,7,8,10,12]]=>[[1,2,4,5,6,7,8,10,12],[3,9,11]]=>20
[1,0,1,1,1,0,0,0,1,1,0,0]=>[[1,3,4,5,9,10],[2,6,7,8,11,12]]=>[[1,2,4,5,6,7,8,11,12],[3,9,10]]=>10
[1,0,1,1,1,0,0,1,0,0,1,0]=>[[1,3,4,5,8,11],[2,6,7,9,10,12]]=>[[1,2,4,5,6,7,9,10,12],[3,8,11]]=>19
[1,0,1,1,1,0,0,1,0,1,0,0]=>[[1,3,4,5,8,10],[2,6,7,9,11,12]]=>[[1,2,4,5,6,7,9,11,12],[3,8,10]]=>18
[1,0,1,1,1,0,0,1,1,0,0,0]=>[[1,3,4,5,8,9],[2,6,7,10,11,12]]=>[[1,2,4,5,6,7,10,11,12],[3,8,9]]=>9
[1,0,1,1,1,0,1,0,0,0,1,0]=>[[1,3,4,5,7,11],[2,6,8,9,10,12]]=>[[1,2,4,5,6,8,9,10,12],[3,7,11]]=>18
[1,0,1,1,1,0,1,0,0,1,0,0]=>[[1,3,4,5,7,10],[2,6,8,9,11,12]]=>[[1,2,4,5,6,8,9,11,12],[3,7,10]]=>17
[1,0,1,1,1,0,1,0,1,0,0,0]=>[[1,3,4,5,7,9],[2,6,8,10,11,12]]=>[[1,2,4,5,6,8,10,11,12],[3,7,9]]=>16
[1,0,1,1,1,0,1,1,0,0,0,0]=>[[1,3,4,5,7,8],[2,6,9,10,11,12]]=>[[1,2,4,5,6,8,9,10,11,12],[3,7]]=>8
[1,0,1,1,1,1,0,0,0,0,1,0]=>[[1,3,4,5,6,11],[2,7,8,9,10,12]]=>[[1,2,4,5,6,7,8,9,10,12],[3,11]]=>12
[1,0,1,1,1,1,0,0,0,1,0,0]=>[[1,3,4,5,6,10],[2,7,8,9,11,12]]=>[[1,2,4,5,6,7,8,9,11,12],[3,10]]=>11
[1,0,1,1,1,1,0,0,1,0,0,0]=>[[1,3,4,5,6,9],[2,7,8,10,11,12]]=>[[1,2,4,5,6,7,8,10,11,12],[3,9]]=>10
[1,0,1,1,1,1,0,1,0,0,0,0]=>[[1,3,4,5,6,8],[2,7,9,10,11,12]]=>[[1,2,4,5,6,7,9,10,11,12],[3,8]]=>9
[1,0,1,1,1,1,1,0,0,0,0,0]=>[[1,3,4,5,6,7],[2,8,9,10,11,12]]=>[[1,2,4,5,6,7,8,9,10,11,12],[3]]=>2
[1,1,0,0,1,0,1,0,1,0,1,0]=>[[1,2,5,7,9,11],[3,4,6,8,10,12]]=>[[1,2,3,4,6,8,10,12],[5,7,9,11]]=>28
[1,1,0,0,1,0,1,0,1,1,0,0]=>[[1,2,5,7,9,10],[3,4,6,8,11,12]]=>[[1,2,3,4,6,8,11,12],[5,7,9,10]]=>18
[1,1,0,0,1,0,1,1,0,0,1,0]=>[[1,2,5,7,8,11],[3,4,6,9,10,12]]=>[[1,2,3,4,6,9,10,12],[5,7,8,11]]=>20
[1,1,0,0,1,0,1,1,0,1,0,0]=>[[1,2,5,7,8,10],[3,4,6,9,11,12]]=>[[1,2,3,4,6,9,11,12],[5,7,8,10]]=>19
[1,1,0,0,1,0,1,1,1,0,0,0]=>[[1,2,5,7,8,9],[3,4,6,10,11,12]]=>[[1,2,3,4,6,9,10,11,12],[5,7,8]]=>10
[1,1,0,0,1,1,0,0,1,0,1,0]=>[[1,2,5,6,9,11],[3,4,7,8,10,12]]=>[[1,2,3,4,7,8,10,12],[5,6,9,11]]=>22
[1,1,0,0,1,1,0,0,1,1,0,0]=>[[1,2,5,6,9,10],[3,4,7,8,11,12]]=>[[1,2,3,4,7,8,11,12],[5,6,9,10]]=>12
[1,1,0,0,1,1,0,1,0,0,1,0]=>[[1,2,5,6,8,11],[3,4,7,9,10,12]]=>[[1,2,3,4,7,9,10,12],[5,6,8,11]]=>21
[1,1,0,0,1,1,0,1,0,1,0,0]=>[[1,2,5,6,8,10],[3,4,7,9,11,12]]=>[[1,2,3,4,7,9,11,12],[5,6,8,10]]=>20
[1,1,0,0,1,1,0,1,1,0,0,0]=>[[1,2,5,6,8,9],[3,4,7,10,11,12]]=>[[1,2,3,4,7,9,10,11,12],[5,6,8]]=>11
[1,1,0,0,1,1,1,0,0,0,1,0]=>[[1,2,5,6,7,11],[3,4,8,9,10,12]]=>[[1,2,3,4,7,8,9,10,12],[5,6,11]]=>14
[1,1,0,0,1,1,1,0,0,1,0,0]=>[[1,2,5,6,7,10],[3,4,8,9,11,12]]=>[[1,2,3,4,7,8,9,11,12],[5,6,10]]=>13
[1,1,0,0,1,1,1,0,1,0,0,0]=>[[1,2,5,6,7,9],[3,4,8,10,11,12]]=>[[1,2,3,4,7,8,10,11,12],[5,6,9]]=>12
[1,1,0,0,1,1,1,1,0,0,0,0]=>[[1,2,5,6,7,8],[3,4,9,10,11,12]]=>[[1,2,3,4,7,8,9,10,11,12],[5,6]]=>4
[1,1,0,1,0,0,1,0,1,0,1,0]=>[[1,2,4,7,9,11],[3,5,6,8,10,12]]=>[[1,2,3,5,6,8,10,12],[4,7,9,11]]=>27
[1,1,0,1,0,0,1,0,1,1,0,0]=>[[1,2,4,7,9,10],[3,5,6,8,11,12]]=>[[1,2,3,5,6,8,11,12],[4,7,9,10]]=>17
[1,1,0,1,0,0,1,1,0,0,1,0]=>[[1,2,4,7,8,11],[3,5,6,9,10,12]]=>[[1,2,3,5,6,9,10,12],[4,7,8,11]]=>19
[1,1,0,1,0,0,1,1,0,1,0,0]=>[[1,2,4,7,8,10],[3,5,6,9,11,12]]=>[[1,2,3,5,6,9,11,12],[4,7,8,10]]=>18
[1,1,0,1,0,0,1,1,1,0,0,0]=>[[1,2,4,7,8,9],[3,5,6,10,11,12]]=>[[1,2,3,5,6,9,10,11,12],[4,7,8]]=>9
[1,1,0,1,0,1,0,0,1,0,1,0]=>[[1,2,4,6,9,11],[3,5,7,8,10,12]]=>[[1,2,3,5,7,8,10,12],[4,6,9,11]]=>26
[1,1,0,1,0,1,0,0,1,1,0,0]=>[[1,2,4,6,9,10],[3,5,7,8,11,12]]=>[[1,2,3,5,7,8,11,12],[4,6,9,10]]=>16
[1,1,0,1,0,1,0,1,0,0,1,0]=>[[1,2,4,6,8,11],[3,5,7,9,10,12]]=>[[1,2,3,5,7,9,10,12],[4,6,8,11]]=>25
[1,1,0,1,0,1,0,1,0,1,0,0]=>[[1,2,4,6,8,10],[3,5,7,9,11,12]]=>[[1,2,3,5,7,9,11,12],[4,6,8,10]]=>24
[1,1,0,1,0,1,0,1,1,0,0,0]=>[[1,2,4,6,8,9],[3,5,7,10,11,12]]=>[[1,2,3,5,7,9,10,11,12],[4,6,8]]=>15
[1,1,0,1,0,1,1,0,0,0,1,0]=>[[1,2,4,6,7,11],[3,5,8,9,10,12]]=>[[1,2,3,5,7,8,9,10,12],[4,6,11]]=>18
[1,1,0,1,0,1,1,0,0,1,0,0]=>[[1,2,4,6,7,10],[3,5,8,9,11,12]]=>[[1,2,3,5,7,8,9,11,12],[4,6,10]]=>17
[1,1,0,1,0,1,1,0,1,0,0,0]=>[[1,2,4,6,7,9],[3,5,8,10,11,12]]=>[[1,2,3,5,7,8,10,11,12],[4,6,9]]=>16
[1,1,0,1,0,1,1,1,0,0,0,0]=>[[1,2,4,6,7,8],[3,5,9,10,11,12]]=>[[1,2,3,5,7,8,9,10,11,12],[4,6]]=>8
[1,1,0,1,1,0,0,0,1,0,1,0]=>[[1,2,4,5,9,11],[3,6,7,8,10,12]]=>[[1,2,3,5,6,7,8,10,12],[4,9,11]]=>21
[1,1,0,1,1,0,0,0,1,1,0,0]=>[[1,2,4,5,9,10],[3,6,7,8,11,12]]=>[[1,2,3,5,6,7,8,11,12],[4,9,10]]=>11
[1,1,0,1,1,0,0,1,0,0,1,0]=>[[1,2,4,5,8,11],[3,6,7,9,10,12]]=>[[1,2,3,5,6,7,9,10,12],[4,8,11]]=>20
[1,1,0,1,1,0,0,1,0,1,0,0]=>[[1,2,4,5,8,10],[3,6,7,9,11,12]]=>[[1,2,3,5,6,7,9,11,12],[4,8,10]]=>19
[1,1,0,1,1,0,0,1,1,0,0,0]=>[[1,2,4,5,8,9],[3,6,7,10,11,12]]=>[[1,2,3,5,6,7,10,11,12],[4,8,9]]=>10
[1,1,0,1,1,0,1,0,0,0,1,0]=>[[1,2,4,5,7,11],[3,6,8,9,10,12]]=>[[1,2,3,5,6,8,9,10,12],[4,7,11]]=>19
[1,1,0,1,1,0,1,0,0,1,0,0]=>[[1,2,4,5,7,10],[3,6,8,9,11,12]]=>[[1,2,3,5,6,8,9,11,12],[4,7,10]]=>18
[1,1,0,1,1,0,1,0,1,0,0,0]=>[[1,2,4,5,7,9],[3,6,8,10,11,12]]=>[[1,2,3,5,6,8,10,11,12],[4,7,9]]=>17
[1,1,0,1,1,0,1,1,0,0,0,0]=>[[1,2,4,5,7,8],[3,6,9,10,11,12]]=>[[1,2,3,5,6,8,9,10,11,12],[4,7]]=>9
[1,1,0,1,1,1,0,0,0,0,1,0]=>[[1,2,4,5,6,11],[3,7,8,9,10,12]]=>[[1,2,3,5,6,7,8,9,10,12],[4,11]]=>13
[1,1,0,1,1,1,0,0,0,1,0,0]=>[[1,2,4,5,6,10],[3,7,8,9,11,12]]=>[[1,2,3,5,6,7,8,9,11,12],[4,10]]=>12
[1,1,0,1,1,1,0,0,1,0,0,0]=>[[1,2,4,5,6,9],[3,7,8,10,11,12]]=>[[1,2,3,5,6,7,8,10,11,12],[4,9]]=>11
[1,1,0,1,1,1,0,1,0,0,0,0]=>[[1,2,4,5,6,8],[3,7,9,10,11,12]]=>[[1,2,3,5,6,7,9,10,11,12],[4,8]]=>10
[1,1,0,1,1,1,1,0,0,0,0,0]=>[[1,2,4,5,6,7],[3,8,9,10,11,12]]=>[[1,2,3,5,6,7,8,9,10,11,12],[4]]=>3
[1,1,1,0,0,0,1,0,1,0,1,0]=>[[1,2,3,7,9,11],[4,5,6,8,10,12]]=>[[1,2,3,4,5,6,8,10,12],[7,9,11]]=>24
[1,1,1,0,0,0,1,0,1,1,0,0]=>[[1,2,3,7,9,10],[4,5,6,8,11,12]]=>[[1,2,3,4,5,6,8,11,12],[7,9,10]]=>14
[1,1,1,0,0,0,1,1,0,0,1,0]=>[[1,2,3,7,8,11],[4,5,6,9,10,12]]=>[[1,2,3,4,5,6,9,10,12],[7,8,11]]=>16
[1,1,1,0,0,0,1,1,0,1,0,0]=>[[1,2,3,7,8,10],[4,5,6,9,11,12]]=>[[1,2,3,4,5,6,9,11,12],[7,8,10]]=>15
[1,1,1,0,0,0,1,1,1,0,0,0]=>[[1,2,3,7,8,9],[4,5,6,10,11,12]]=>[[1,2,3,4,5,6,10,11,12],[7,8,9]]=>6
[1,1,1,0,0,1,0,0,1,0,1,0]=>[[1,2,3,6,9,11],[4,5,7,8,10,12]]=>[[1,2,3,4,5,7,8,10,12],[6,9,11]]=>23
[1,1,1,0,0,1,0,0,1,1,0,0]=>[[1,2,3,6,9,10],[4,5,7,8,11,12]]=>[[1,2,3,4,5,7,8,11,12],[6,9,10]]=>13
[1,1,1,0,0,1,0,1,0,0,1,0]=>[[1,2,3,6,8,11],[4,5,7,9,10,12]]=>[[1,2,3,4,5,7,9,10,12],[6,8,11]]=>22
[1,1,1,0,0,1,0,1,0,1,0,0]=>[[1,2,3,6,8,10],[4,5,7,9,11,12]]=>[[1,2,3,4,5,7,9,11,12],[6,8,10]]=>21
[1,1,1,0,0,1,0,1,1,0,0,0]=>[[1,2,3,6,8,9],[4,5,7,10,11,12]]=>[[1,2,3,4,5,7,10,11,12],[6,8,9]]=>12
[1,1,1,0,0,1,1,0,0,0,1,0]=>[[1,2,3,6,7,11],[4,5,8,9,10,12]]=>[[1,2,3,4,5,8,9,10,12],[6,7,11]]=>15
[1,1,1,0,0,1,1,0,0,1,0,0]=>[[1,2,3,6,7,10],[4,5,8,9,11,12]]=>[[1,2,3,4,5,8,9,11,12],[6,7,10]]=>14
[1,1,1,0,0,1,1,0,1,0,0,0]=>[[1,2,3,6,7,9],[4,5,8,10,11,12]]=>[[1,2,3,4,5,8,10,11,12],[6,7,9]]=>13
[1,1,1,0,0,1,1,1,0,0,0,0]=>[[1,2,3,6,7,8],[4,5,9,10,11,12]]=>[[1,2,3,4,5,8,9,10,11,12],[6,7]]=>5
[1,1,1,0,1,0,0,0,1,0,1,0]=>[[1,2,3,5,9,11],[4,6,7,8,10,12]]=>[[1,2,3,4,6,7,8,10,12],[5,9,11]]=>22
[1,1,1,0,1,0,0,0,1,1,0,0]=>[[1,2,3,5,9,10],[4,6,7,8,11,12]]=>[[1,2,3,4,6,7,8,11,12],[5,9,10]]=>12
[1,1,1,0,1,0,0,1,0,0,1,0]=>[[1,2,3,5,8,11],[4,6,7,9,10,12]]=>[[1,2,3,4,6,7,9,10,12],[5,8,11]]=>21
[1,1,1,0,1,0,0,1,0,1,0,0]=>[[1,2,3,5,8,10],[4,6,7,9,11,12]]=>[[1,2,3,4,6,7,9,11,12],[5,8,10]]=>20
[1,1,1,0,1,0,0,1,1,0,0,0]=>[[1,2,3,5,8,9],[4,6,7,10,11,12]]=>[[1,2,3,4,6,7,10,11,12],[5,8,9]]=>11
[1,1,1,0,1,0,1,0,0,0,1,0]=>[[1,2,3,5,7,11],[4,6,8,9,10,12]]=>[[1,2,3,4,6,8,9,10,12],[5,7,11]]=>20
[1,1,1,0,1,0,1,0,0,1,0,0]=>[[1,2,3,5,7,10],[4,6,8,9,11,12]]=>[[1,2,3,4,6,8,9,11,12],[5,7,10]]=>19
[1,1,1,0,1,0,1,0,1,0,0,0]=>[[1,2,3,5,7,9],[4,6,8,10,11,12]]=>[[1,2,3,4,6,8,10,11,12],[5,7,9]]=>18
[1,1,1,0,1,0,1,1,0,0,0,0]=>[[1,2,3,5,7,8],[4,6,9,10,11,12]]=>[[1,2,3,4,6,8,9,10,11,12],[5,7]]=>10
[1,1,1,0,1,1,0,0,0,0,1,0]=>[[1,2,3,5,6,11],[4,7,8,9,10,12]]=>[[1,2,3,4,6,7,8,9,10,12],[5,11]]=>14
[1,1,1,0,1,1,0,0,0,1,0,0]=>[[1,2,3,5,6,10],[4,7,8,9,11,12]]=>[[1,2,3,4,6,7,8,9,11,12],[5,10]]=>13
[1,1,1,0,1,1,0,0,1,0,0,0]=>[[1,2,3,5,6,9],[4,7,8,10,11,12]]=>[[1,2,3,4,6,7,8,10,11,12],[5,9]]=>12
[1,1,1,0,1,1,0,1,0,0,0,0]=>[[1,2,3,5,6,8],[4,7,9,10,11,12]]=>[[1,2,3,4,6,7,9,10,11,12],[5,8]]=>11
[1,1,1,0,1,1,1,0,0,0,0,0]=>[[1,2,3,5,6,7],[4,8,9,10,11,12]]=>[[1,2,3,4,6,7,8,9,10,11,12],[5]]=>4
[1,1,1,1,0,0,0,0,1,0,1,0]=>[[1,2,3,4,9,11],[5,6,7,8,10,12]]=>[[1,2,3,4,5,6,7,8,10,12],[9,11]]=>18
[1,1,1,1,0,0,0,0,1,1,0,0]=>[[1,2,3,4,9,10],[5,6,7,8,11,12]]=>[[1,2,3,4,5,6,7,8,11,12],[9,10]]=>8
[1,1,1,1,0,0,0,1,0,0,1,0]=>[[1,2,3,4,8,11],[5,6,7,9,10,12]]=>[[1,2,3,4,5,6,7,9,10,12],[8,11]]=>17
[1,1,1,1,0,0,0,1,0,1,0,0]=>[[1,2,3,4,8,10],[5,6,7,9,11,12]]=>[[1,2,3,4,5,6,7,9,11,12],[8,10]]=>16
[1,1,1,1,0,0,0,1,1,0,0,0]=>[[1,2,3,4,8,9],[5,6,7,10,11,12]]=>[[1,2,3,4,5,6,7,10,11,12],[8,9]]=>7
[1,1,1,1,0,0,1,0,0,0,1,0]=>[[1,2,3,4,7,11],[5,6,8,9,10,12]]=>[[1,2,3,4,5,6,8,9,10,12],[7,11]]=>16
[1,1,1,1,0,0,1,0,0,1,0,0]=>[[1,2,3,4,7,10],[5,6,8,9,11,12]]=>[[1,2,3,4,5,6,8,9,11,12],[7,10]]=>15
[1,1,1,1,0,0,1,0,1,0,0,0]=>[[1,2,3,4,7,9],[5,6,8,10,11,12]]=>[[1,2,3,4,5,6,8,10,11,12],[7,9]]=>14
[1,1,1,1,0,0,1,1,0,0,0,0]=>[[1,2,3,4,7,8],[5,6,9,10,11,12]]=>[[1,2,3,4,5,6,9,10,11,12],[7,8]]=>6
[1,1,1,1,0,1,0,0,0,0,1,0]=>[[1,2,3,4,6,11],[5,7,8,9,10,12]]=>[[1,2,3,4,5,7,8,9,10,12],[6,11]]=>15
[1,1,1,1,0,1,0,0,0,1,0,0]=>[[1,2,3,4,6,10],[5,7,8,9,11,12]]=>[[1,2,3,4,5,7,8,9,11,12],[6,10]]=>14
[1,1,1,1,0,1,0,0,1,0,0,0]=>[[1,2,3,4,6,9],[5,7,8,10,11,12]]=>[[1,2,3,4,5,7,8,10,11,12],[6,9]]=>13
[1,1,1,1,0,1,0,1,0,0,0,0]=>[[1,2,3,4,6,8],[5,7,9,10,11,12]]=>[[1,2,3,4,5,7,9,10,11,12],[6,8]]=>12
[1,1,1,1,0,1,1,0,0,0,0,0]=>[[1,2,3,4,6,7],[5,8,9,10,11,12]]=>[[1,2,3,4,5,7,8,9,10,11,12],[6]]=>5
[1,1,1,1,1,0,0,0,0,0,1,0]=>[[1,2,3,4,5,11],[6,7,8,9,10,12]]=>[[1,2,3,4,5,6,7,8,9,10,12],[11]]=>10
[1,1,1,1,1,0,0,0,0,1,0,0]=>[[1,2,3,4,5,10],[6,7,8,9,11,12]]=>[[1,2,3,4,5,6,7,8,9,11,12],[10]]=>9
[1,1,1,1,1,0,0,0,1,0,0,0]=>[[1,2,3,4,5,9],[6,7,8,10,11,12]]=>[[1,2,3,4,5,6,7,8,10,11,12],[9]]=>8
[1,1,1,1,1,0,0,1,0,0,0,0]=>[[1,2,3,4,5,8],[6,7,9,10,11,12]]=>[[1,2,3,4,5,6,7,9,10,11,12],[8]]=>7
[1,1,1,1,1,0,1,0,0,0,0,0]=>[[1,2,3,4,5,7],[6,8,9,10,11,12]]=>[[1,2,3,4,5,6,8,9,10,11,12],[7]]=>6
[1,1,1,1,1,1,0,0,0,0,0,0]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[[1,2,3,4,5,6,7,8,9,10,11,12]]=>0
                    

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$F_{1} = 1$

$F_{2} = 1 + q^{2}$

$F_{3} = 1 + q^{2} + q^{3} + q^{4} + q^{6}$

$F_{4} = 1 + q^{2} + q^{3} + 2\ q^{4} + q^{5} + 2\ q^{6} + q^{7} + 2\ q^{8} + q^{9} + q^{10} + q^{12}$

$F_{5} = 1 + q^{2} + q^{3} + 2\ q^{4} + 2\ q^{5} + 3\ q^{6} + 2\ q^{7} + 4\ q^{8} + 3\ q^{9} + 4\ q^{10} + 3\ q^{11} + 4\ q^{12} + 2\ q^{13} + 3\ q^{14} + 2\ q^{15} + 2\ q^{16} + q^{17} + q^{18} + q^{20}$

$F_{6} = 1 + q^{2} + q^{3} + 2\ q^{4} + 2\ q^{5} + 4\ q^{6} + 3\ q^{7} + 5\ q^{8} + 5\ q^{9} + 7\ q^{10} + 6\ q^{11} + 9\ q^{12} + 7\ q^{13} + 9\ q^{14} + 8\ q^{15} + 9\ q^{16} + 7\ q^{17} + 9\ q^{18} + 6\ q^{19} + 7\ q^{20} + 5\ q^{21} + 5\ q^{22} + 3\ q^{23} + 4\ q^{24} + 2\ q^{25} + 2\ q^{26} + q^{27} + q^{28} + q^{30}$

Description

The (standard) major index of a standard tableau.
A descent of a standard tableau

$T$ is an index

$i$ such that

$i+1$ appears in a row strictly below the row of

$i$ . The (standard) major index is the the sum of the descents.

Map

to two-row standard tableau

Description

Return a standard tableau of shape

$(n,n)$ where

$n$ is the semilength of the Dyck path.
Given a Dyck path

$D$ , its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.

Map

catabolism

Description

Remove the first row of the standard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry

$k$ into tableau

$T$ is as follows:
If

$k$ is larger than all entries in the first column, place

$k$ at the bottom of the first column and the procedure is finished. Otherwise, place

$k$ in the first column, replacing the smallest entry,

$y$ , greater than

$k$ . Now insert

$y$ into the second column using the same procedure: if

$y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place

$y$ in the second column, replacing, or 'bumping', the smallest entry,

$z$ , larger than

$y$ . Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).