Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000872
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2,1] => 0
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,4,3,2] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,3,1] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,4,3] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [1,5,4,3,2] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,4,2] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,4,3,2] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,4,1,3] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [1,6,5,4,3,2] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [4,1,5,3,2] => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,4,1,3] => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,5,1,4,3] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [2,1,6,5,4,3] => 0
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => [1,5,4,3,2] => 0
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,5,4,2] => 0
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [1,5,4,3,2] => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,4,3,1] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,5,1,4,3] => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => [2,6,1,5,4,3] => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,3,6,4] => [1,6,5,4,3,2] => 0
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,6,5] => [4,1,6,5,3,2] => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => 0
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,5,1,4,2] => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => 0
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,1,5,4,2] => 0
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,4,3,2] => 0
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,5,4,1,3] => 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,6,3,4,5] => [2,1,6,5,4,3] => 0
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => [2,6,1,5,4,3] => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => [1,6,5,4,3,2] => 0
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,5,1,2,6,3] => [4,6,1,5,3,2] => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,6,5] => [3,1,6,5,4,2] => 0
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => [1,6,5,4,3,2] => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,1,5,4,2] => 0
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => 0
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,5,1,4,3] => 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => [2,1,6,5,4,3] => 0
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,5,4,1,3] => 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => [2,6,5,1,4,3] => 1
Description
The number of very big descents of a permutation. A very big descent of a permutation $\pi$ is an index $i$ such that $\pi_i - \pi_{i+1} > 2$. For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General $r$-descents were for example be studied in [1, Section 2].
Matching statistic: St000455
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 0
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 0
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 0
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 0
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => 0
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,3]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[5,2,1]
 => 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,1,1,1]
 => 100001110 => [1,5,1,1,2] => ([(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0
[4,4]
 => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 0
[4,3,1]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[4,2,2]
 => 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[4,2,1,1]
 => 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 0
[3,3,2]
 => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,3,1,1]
 => 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[3,2,2,1]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,2,2,2]
 => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,4]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[5,3,1]
 => 10010010 => [1,3,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[5,2,2]
 => 10001100 => [1,4,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[5,2,1,1]
 => 100010110 => [1,4,2,1,2] => ([(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[5,1,1,1,1]
 => 1000011110 => [1,5,1,1,1,2] => ([(1,7),(1,8),(1,9),(1,10),(2,7),(2,8),(2,9),(2,10),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
 => ? = 0
[4,4,1]
 => 1100010 => [1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,3,2]
 => 1010100 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[4,3,1,1]
 => 10100110 => [1,2,3,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[4,2,2,1]
 => 10011010 => [1,3,1,2,2] => ([(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[4,2,1,1,1]
 => 100101110 => [1,3,2,1,1,2] => ([(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[3,3,3]
 => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[3,3,2,1]
 => 1101010 => [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,3,1,1,1]
 => 11001110 => [1,1,3,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000629
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 1010 => 0010 => 0
[2]
 => [1,1,0,0,1,0]
 => 110010 => 010010 => 0
[1,1]
 => [1,0,1,1,0,0]
 => 101100 => 001100 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 01100010 => 0
[2,1]
 => [1,0,1,0,1,0]
 => 101010 => 001010 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 00111000 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0111000010 => ? = 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 01010010 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 01001100 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 00110100 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0011110000 => ? = 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => 011110000010 => ? = 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0110100010 => ? = 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 01001010 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 00110010 => 0
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 00101100 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0011101000 => ? = 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 001111100000 => ? = 0
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 111101000010 => 011101000010 => ? = 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0110010010 => ? = 0
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0101100010 => ? = 0
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0110001100 => ? = 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 00101010 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0011100100 => ? = 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0100111000 => ? = 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0011011000 => ? = 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => 001111010000 => ? = 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 111100100010 => 011100100010 => ? = 0
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 111011000010 => 011011000010 => ? = 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0110001010 => ? = 0
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0101010010 => ? = 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0011100010 => ? = 0
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0101001100 => ? = 0
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0100110100 => ? = 0
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0011010100 => ? = 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 101111001000 => 001111001000 => ? = 0
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0010111000 => ? = 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 101110110000 => 001110110000 => ? = 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 111100010010 => 011100010010 => ? = 0
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 111010100010 => 011010100010 => ? = 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 110111000010 => 010111000010 => ? = 0
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 111100001100 => 011100001100 => ? = 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0101001010 => ? = 0
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0100110010 => ? = 0
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0011010010 => ? = 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 101111000100 => 001111000100 => ? = 0
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0100101100 => ? = 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0011001100 => ? = 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0010110100 => ? = 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 101110101000 => 001110101000 => ? = 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 010011110000 => ? = 0
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 101101110000 => 001101110000 => ? = 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 111100001010 => 011100001010 => ? = 0
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 111010010010 => 011010010010 => ? = 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 111001100010 => 011001100010 => ? = 0
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 110110100010 => 010110100010 => ? = 1
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 101111000010 => 001111000010 => ? = 0
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => 111010001100 => 011010001100 => ? = 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0100101010 => ? = 0
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0011001010 => ? = 0
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0010110010 => ? = 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 101110100100 => 001110100100 => ? = 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 011000111000 => ? = 0
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0010101100 => 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0010101010 => 0
Description
The defect of a binary word. The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St001811
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,1,0,0]
 => [1,2] => 0
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => 0
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 0
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => 0
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,1,2] => ? = 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,1,2,6] => ? = 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,6,1,2,5] => ? = 0
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,5,6,1,2,4] => ? = 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,1,2,3] => ? = 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,1,2,7] => ? = 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,1,2,5,6] => ? = 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,5,7,1,2,6] => ? = 0
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,6,2,4,5] => ? = 0
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,4,6,7,1,2,5] => ? = 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,5,6,2,3,4] => ? = 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,5,6,7,1,2,4] => ? = 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6,7] => ? = 0
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,5,6,8,1,2,7] => ? = 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,6,1,2,4,5] => ? = 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [3,4,1,7,2,5,6] => ? = 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,4,5,7,8,1,2,6] => ? = 0
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [5,1,6,2,3,4] => ? = 0
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,1,2,4,5,6] => ? = 0
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,6,7,2,4,5] => ? = 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,6,7,8,1,2,5] => ? = 0
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5] => ? = 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,5,6,7,2,3,4] => ? = 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [3,4,7,1,2,5,6] => ? = 0
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [3,4,5,1,8,2,6,7] => ? = 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,4,5,6,8,9,1,2,7] => ? = 0
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,6,1,2,3,4] => ? = 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [3,6,1,7,2,4,5] => ? = 0
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => ? = 0
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [3,4,1,7,8,2,5,6] => ? = 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,7,8,9,1,2,6] => ? = 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [5,1,2,3,4,6] => ? = 0
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,1,6,7,2,3,4] => ? = 0
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [3,1,6,7,8,2,4,5] => ? = 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,1,2,3,4,5] => ? = 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,2,6,7,3,4,5] => ? = 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [3,6,7,1,2,4,5] => ? = 0
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [3,4,7,1,8,2,5,6] => ? = 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,1,2,6,7,8] => ? = 0
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [3,4,5,1,8,9,2,6,7] => ? = 1
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,8,9,10,1,2,7] => ? = 0
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,6,1,7,2,3,4] => ? = 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5,7] => ? = 0
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [3,6,1,7,8,2,4,5] => ? = 0
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [3,4,1,2,8,5,6,7] => ? = 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [3,4,1,7,8,9,2,5,6] => ? = 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ? = 0
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [5,1,2,7,3,4,6] => ? = 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
 => [5,1,6,7,8,2,3,4] => ? = 0
Description
The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001208
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [2,1] => [1,2] => 1 = 0 + 1
[2]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 1 = 0 + 1
[1,1]
 => [1,1,0,0]
 => [2,3,1] => [3,1,2] => 1 = 0 + 1
[3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,3,4,2] => 1 = 0 + 1
[2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => 1 = 0 + 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,2,4,3] => 1 = 0 + 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,3,4,5,2] => 1 = 0 + 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,2,3,1,4] => 1 = 0 + 1
[2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [3,4,1,2] => 1 = 0 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,3,2,5,4] => 2 = 1 + 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,4,5,3] => 1 = 0 + 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,3,4,5,6,2] => ? = 0 + 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,2,3,4,1,5] => ? = 1 + 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [4,2,5,1,3] => 1 = 0 + 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,3,4,2,6,5] => ? = 0 + 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [3,1,2,5,4] => 2 = 1 + 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,3,2,5,6,4] => ? = 1 + 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [6,1,3,4,5,2] => ? = 0 + 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [7,2,3,4,5,1,6] => ? = 1 + 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [5,2,3,6,1,4] => ? = 0 + 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => [1,3,4,5,2,7,6] => ? = 0 + 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => 1 = 0 + 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,2,1,3,6,5] => ? = 0 + 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [1,3,4,2,6,7,5] => ? = 0 + 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => 1 = 0 + 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [3,1,2,5,6,4] => ? = 1 + 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [7,2,1,4,5,6,3] => ? = 1 + 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => [6,2,3,4,7,1,5] => ? = 0 + 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [8,1,2,3,4,7,5,6] => [1,3,4,5,6,2,8,7] => ? = 1 + 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,3,6,2,5,4] => ? = 0 + 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => [5,2,3,1,4,7,6] => ? = 1 + 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => [1,3,4,5,2,7,8,6] => ? = 0 + 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [1,5,2,4,6,3] => ? = 0 + 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [4,2,5,6,1,3] => ? = 0 + 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [4,2,1,3,6,7,5] => ? = 1 + 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => [8,2,3,1,5,6,7,4] => ? = 0 + 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [3,4,1,2,6,5] => ? = 1 + 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [3,7,1,4,5,6,2] => ? = 1 + 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => [1,3,4,7,2,6,5] => ? = 0 + 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,1,2,3,8,7,4,6] => [6,2,3,4,1,5,8,7] => ? = 1 + 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [9,1,2,3,4,8,5,6,7] => [1,3,4,5,6,2,8,9,7] => ? = 0 + 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [1,2,3,5,6,4] => ? = 0 + 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [1,3,6,2,5,7,4] => ? = 0 + 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [5,2,3,6,7,1,4] => ? = 0 + 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => [5,2,3,1,4,7,8,6] => ? = 1 + 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [8,1,2,3,9,4,5,6,7] => [9,2,3,4,1,6,7,8,5] => ? = 0 + 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [6,4,5,2,1,3] => ? = 0 + 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [7,4,1,3,5,6,2] => ? = 0 + 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => [4,2,5,1,3,7,6] => ? = 1 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,7,8,2,4,5,6] => [4,2,8,1,5,6,7,3] => ? = 1 + 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [1,5,6,2,4,3] => ? = 0 + 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [3,4,1,2,6,7,5] => ? = 1 + 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [1,3,2,4,6,7,5] => ? = 0 + 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [8,1,2,5,7,3,4,6] => [1,3,4,7,2,6,8,5] => ? = 1 + 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,1,2,3,6,7,8,4] => [6,2,3,4,7,8,1,5] => ? = 0 + 1
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,1,2,3,9,8,4,6,7] => [6,2,3,4,1,5,8,9,7] => ? = 1 + 1
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [9,1,2,3,4,10,5,6,7,8] => [10,2,3,4,5,1,7,8,9,6] => ? = 0 + 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [7,1,6,3,4,5,2] => ? = 1 + 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [7,2,5,6,3,1,4] => ? = 0 + 1
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [8,2,5,1,4,6,7,3] => ? = 0 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,8,7,3,6] => [5,2,3,6,1,4,8,7] => ? = 1 + 1
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,1,2,8,9,3,5,6,7] => [5,2,3,9,1,6,7,8,4] => ? = 1 + 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [3,4,5,6,1,2] => ? = 0 + 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => [1,5,6,3,2,7,4] => ? = 1 + 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [8,3,7,1,2,4,5,6] => [1,5,2,4,6,7,8,3] => ? = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St000955
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000955: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,1]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[3]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,1,1]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[4]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 0 + 2
[3,1]
 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,2]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,1]
 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[5]
 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 2
[4,1]
 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1 + 2
[3,2]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[3,1,1]
 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 0 + 2
[2,2,1]
 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[5,1]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1 + 2
[4,2]
 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 0 + 2
[4,1,1]
 => 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 0 + 2
[3,3]
 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[3,2,1]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 2
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[2,2,2]
 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[5,2]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 0 + 2
[5,1,1]
 => 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[4,3]
 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 2
[4,2,1]
 => 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[3,3,1]
 => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0 + 2
[3,2,2]
 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 2
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[5,3]
 => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 2
[5,2,1]
 => 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[5,1,1,1]
 => 100001110 => [1,5,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[4,4]
 => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[4,3,1]
 => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0 + 2
[4,2,2]
 => 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 2
[4,2,1,1]
 => 10010110 => [1,3,2,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[4,1,1,1,1]
 => 100011110 => [1,4,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[3,3,2]
 => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 2
[3,3,1,1]
 => 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 0 + 2
[3,2,2,1]
 => 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[3,2,1,1,1]
 => 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[2,2,2,2]
 => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 2
[2,2,2,1,1]
 => 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[5,4]
 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 2
[5,3,1]
 => 10010010 => [1,3,3,2] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1 + 2
[5,2,2]
 => 10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 2
[5,2,1,1]
 => 100010110 => [1,4,2,1,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 2
[5,1,1,1,1]
 => 1000011110 => [1,5,1,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[4,4,1]
 => 1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1 + 2
[4,3,2]
 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 2
[4,3,1,1]
 => 10100110 => [1,2,3,1,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 0 + 2
[4,2,2,1]
 => 10011010 => [1,3,1,2,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[4,2,1,1,1]
 => 100101110 => [1,3,2,1,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 2
[3,3,3]
 => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 2
[3,3,2,1]
 => 1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 2
[3,3,1,1,1]
 => 11001110 => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
Description
Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra.