Identifier
Values
([],3) => [1,1,1] => [1,1] => [1] => 0
([],4) => [1,1,1,1] => [1,1,1] => [1,1] => 0
([(2,3)],4) => [2,1,1] => [1,1] => [1] => 0
([],5) => [1,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(3,4)],5) => [2,1,1,1] => [1,1,1] => [1,1] => 0
([(2,4),(3,4)],5) => [3,1,1] => [1,1] => [1] => 0
([(1,4),(2,3)],5) => [2,2,1] => [2,1] => [1] => 0
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1] => [1] => 0
([],6) => [1,1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1] => 0
([(4,5)],6) => [2,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [1,1] => 0
([(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1] => [1] => 0
([(2,5),(3,4)],6) => [2,2,1,1] => [2,1,1] => [1,1] => 0
([(2,5),(3,4),(4,5)],6) => [4,1,1] => [1,1] => [1] => 0
([(1,2),(3,5),(4,5)],6) => [3,2,1] => [2,1] => [1] => 0
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [1,1] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => [1] => 0
([(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,1] => [1] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => [1] => 0
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [2,2] => [2] => 1
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [2,1] => [1] => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => [1] => 0
([],7) => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,1,1,1,1] => 0
([(5,6)],7) => [2,1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1] => 0
([(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(3,6),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => [1,1] => 0
([(2,6),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(3,6),(4,5)],7) => [2,2,1,1,1] => [2,1,1,1] => [1,1,1] => 0
([(3,6),(4,5),(5,6)],7) => [4,1,1,1] => [1,1,1] => [1,1] => 0
([(2,3),(4,6),(5,6)],7) => [3,2,1,1] => [2,1,1] => [1,1] => 0
([(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(2,6),(3,6),(4,5),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(1,2),(3,6),(4,6),(5,6)],7) => [4,2,1] => [2,1] => [1] => 0
([(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => [1,1] => 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => [1,1,1] => [1,1] => 0
([(1,6),(2,6),(3,5),(4,5)],7) => [3,3,1] => [3,1] => [1] => 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => [1,1] => 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(1,6),(2,5),(3,4)],7) => [2,2,2,1] => [2,2,1] => [2,1] => 1
([(2,6),(3,5),(4,5),(4,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(1,2),(3,6),(4,5),(5,6)],7) => [4,2,1] => [2,1] => [1] => 0
([(0,3),(1,2),(4,6),(5,6)],7) => [3,2,2] => [2,2] => [2] => 1
([(2,3),(4,5),(4,6),(5,6)],7) => [3,2,1,1] => [2,1,1] => [1,1] => 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [2,1] => [1] => 0
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,2,1] => [2,1] => [1] => 0
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,3,1] => [3,1] => [1] => 0
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [2,1] => [1] => 0
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => [1,1] => 0
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) => [3,2,2] => [2,2] => [2] => 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) => [3,3,1] => [3,1] => [1] => 0
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,2,1] => [2,1] => [1] => 0
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => [1] => 0
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(3,7),(4,7),(5,7),(6,7)],8) => [5,1,1,1] => [1,1,1] => [1,1] => 0
([],8) => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
([(4,7),(5,6)],8) => [2,2,1,1,1,1] => [2,1,1,1,1] => [1,1,1,1] => 0
([(4,7),(5,6),(6,7)],8) => [4,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(4,6),(4,7),(5,6),(5,7)],8) => [4,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(2,7),(3,7),(4,6),(5,6)],8) => [3,3,1,1] => [3,1,1] => [1,1] => 0
([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8) => [5,1,1,1] => [1,1,1] => [1,1] => 0
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8) => [3,3,1,1] => [3,1,1] => [1,1] => 0
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,2,1,1] => [2,1,1] => [1,1] => 0
([(2,6),(2,7),(3,4),(3,5),(4,5),(4,7),(5,6),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,3),(2,7),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,3,1] => [3,1] => [1] => 0
([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => [6,1,1] => [1,1] => [1] => 0
([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,3,1] => [3,1] => [1] => 0
([(0,7),(1,6),(2,5),(3,4)],8) => [2,2,2,2] => [2,2,2] => [2,2] => 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8) => [4,2,2] => [2,2] => [2] => 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,2,2] => [2,2] => [2] => 1
([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9) => [7,1,1] => [1,1] => [1] => 0
>>> Load all 132 entries. <<<
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16) => [14,1,1] => [1,1] => [1] => 0
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12) => [10,1,1] => [1,1] => [1] => 0
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,2,2] => [2,2] => [2] => 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9) => [6,1,1,1] => [1,1,1] => [1,1] => 0
([(4,9),(5,8),(6,7),(7,9),(8,9)],10) => [6,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9) => [6,1,1,1] => [1,1,1] => [1,1] => 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12) => [9,1,1,1] => [1,1,1] => [1,1] => 0
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14) => [11,1,1,1] => [1,1,1] => [1,1] => 0
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15) => [11,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10) => [7,1,1,1] => [1,1,1] => [1,1] => 0
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14) => [11,1,1,1] => [1,1,1] => [1,1] => 0
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16) => [13,1,1,1] => [1,1,1] => [1,1] => 0
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11) => [9,1,1] => [1,1] => [1] => 0
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13) => [11,1,1] => [1,1] => [1] => 0
([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10) => [8,1,1] => [1,1] => [1] => 0
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15) => [13,1,1] => [1,1] => [1] => 0
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11) => [8,1,1,1] => [1,1,1] => [1,1] => 0
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15) => [12,1,1,1] => [1,1,1] => [1,1] => 0
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13) => [10,1,1,1] => [1,1,1] => [1,1] => 0
([(5,10),(6,9),(7,8),(8,10),(9,10)],11) => [6,1,1,1,1,1] => [1,1,1,1,1] => [1,1,1,1] => 0
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14) => [12,1,1] => [1,1] => [1] => 0
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11) => [8,1,1,1] => [1,1,1] => [1,1] => 0
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13) => [10,1,1,1] => [1,1,1] => [1,1] => 0
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14) => [10,1,1,1,1] => [1,1,1,1] => [1,1,1] => 0
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14) => [12,1,1] => [1,1] => [1] => 0
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17) => [15,1,1] => [1,1] => [1] => 0
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10) => [7,1,1,1] => [1,1,1] => [1,1] => 0
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14) => [11,1,1,1] => [1,1,1] => [1,1] => 0
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13) => [10,1,1,1] => [1,1,1] => [1,1] => 0
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15) => [13,1,1] => [1,1] => [1] => 0
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12) => [10,1,1] => [1,1] => [1] => 0
search for individual values
searching the database for the individual values of this statistic
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
Map
first row removal
Description
Removes the first entry of an integer partition