Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 2
01 => [1,1] => 2
10 => [1,1] => 2
11 => [2] => 2
000 => [3] => 2
001 => [2,1] => 3
010 => [1,1,1] => 2
011 => [1,2] => 3
100 => [1,2] => 3
101 => [1,1,1] => 2
110 => [2,1] => 3
111 => [3] => 2
0000 => [4] => 2
0001 => [3,1] => 3
0010 => [2,1,1] => 3
0011 => [2,2] => 4
0100 => [1,1,2] => 3
0101 => [1,1,1,1] => 2
0110 => [1,2,1] => 4
0111 => [1,3] => 3
1000 => [1,3] => 3
1001 => [1,2,1] => 4
1010 => [1,1,1,1] => 2
1011 => [1,1,2] => 3
1100 => [2,2] => 4
1101 => [2,1,1] => 3
1110 => [3,1] => 3
1111 => [4] => 2
00000 => [5] => 2
00001 => [4,1] => 3
00010 => [3,1,1] => 3
00011 => [3,2] => 4
00100 => [2,1,2] => 4
00101 => [2,1,1,1] => 3
00110 => [2,2,1] => 5
00111 => [2,3] => 4
01000 => [1,1,3] => 3
01001 => [1,1,2,1] => 4
01010 => [1,1,1,1,1] => 2
01011 => [1,1,1,2] => 3
01100 => [1,2,2] => 5
01101 => [1,2,1,1] => 4
01110 => [1,3,1] => 4
01111 => [1,4] => 3
10000 => [1,4] => 3
10001 => [1,3,1] => 4
10010 => [1,2,1,1] => 4
10011 => [1,2,2] => 5
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St001035
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 70%
Values
0 => [1] => [1,0]
 => [1,0]
 => ? = 1 - 2
1 => [1] => [1,0]
 => [1,0]
 => ? = 1 - 2
00 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 2 - 2
01 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 2 - 2
10 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 2 - 2
11 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 2 - 2
000 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 2 - 2
001 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 2 - 2
011 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 3 - 2
100 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 3 - 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 2 - 2
110 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 3 - 2
111 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 2 - 2
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 3 - 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 2 - 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 2 - 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 3 - 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 3 - 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 4 - 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 3 - 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 5 - 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 4 - 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 3 - 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 4 - 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 2 - 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 3 - 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 4 - 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 4 - 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 3 - 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 4 - 2
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 4 - 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 3 - 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 2 - 2
01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 5 - 2
01001001 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 6 - 2
01001010 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 4 - 2
01001011 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 5 - 2
01001100 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 7 - 2
01001101 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 6 - 2
01001110 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 - 2
01010000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 2
01010001 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 4 - 2
01010010 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 4 - 2
01010011 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 5 - 2
01010100 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 3 - 2
01010101 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 2
01010110 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 4 - 2
01010111 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 3 - 2
01011000 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 - 2
01011001 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6 - 2
01011010 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 2
01011011 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 5 - 2
01011100 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 5 - 2
01011101 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 4 - 2
01011110 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 2
01011111 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 2
10100000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 2
10100001 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 2
10100010 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 4 - 2
10100011 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 5 - 2
10100100 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 5 - 2
10100101 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 4 - 2
10100110 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 6 - 2
10100111 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 - 2
10101000 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 3 - 2
10101001 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 4 - 2
10101010 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 2
10101011 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 3 - 2
10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 5 - 2
10101101 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 4 - 2
10101110 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 4 - 2
10101111 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 2
10110001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6 - 2
10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 6 - 2
10110011 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 7 - 2
10110100 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 5 - 2
10110101 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 4 - 2
10110110 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 6 - 2
10110111 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 5 - 2
000000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 2
000000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 - 2
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000777
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 70%
Values
0 => [1] => ([],1)
 => 1
1 => [1] => ([],1)
 => 1
00 => [2] => ([],2)
 => ? = 2
01 => [1,1] => ([(0,1)],2)
 => 2
10 => [1,1] => ([(0,1)],2)
 => 2
11 => [2] => ([],2)
 => ? = 2
000 => [3] => ([],3)
 => ? = 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 3
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
011 => [1,2] => ([(1,2)],3)
 => ? = 3
100 => [1,2] => ([(1,2)],3)
 => ? = 3
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 3
111 => [3] => ([],3)
 => ? = 2
0000 => [4] => ([],4)
 => ? = 2
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
0111 => [1,3] => ([(2,3)],4)
 => ? = 3
1000 => [1,3] => ([(2,3)],4)
 => ? = 3
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
1100 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1111 => [4] => ([],4)
 => ? = 2
00000 => [5] => ([],5)
 => ? = 2
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
01111 => [1,4] => ([(3,4)],5)
 => ? = 3
10000 => [1,4] => ([(3,4)],5)
 => ? = 3
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3
11000 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
11111 => [5] => ([],5)
 => ? = 2
000000 => [6] => ([],6)
 => ? = 2
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 4
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
000101 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 4
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
001001 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
001010 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6
001101 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
001111 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 3
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
010010 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
010100 => [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)
 => ? = 3
010101 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
010110 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
011010 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
011101 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
011111 => [1,5] => ([(4,5)],6)
 => ? = 3
100000 => [1,5] => ([(4,5)],6)
 => ? = 3
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
100010 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000483
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000483: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 60%
Values
0 => [1] => [1,0]
 => [2,1] => 0 = 1 - 1
1 => [1] => [1,0]
 => [2,1] => 0 = 1 - 1
00 => [2] => [1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
 => [3,1,2] => 1 = 2 - 1
10 => [1,1] => [1,0,1,0]
 => [3,1,2] => 1 = 2 - 1
11 => [2] => [1,1,0,0]
 => [2,3,1] => 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2 = 3 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2 = 3 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 2 = 3 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 1 = 2 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2 = 3 - 1
111 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 2 = 3 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 2 = 3 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 3 = 4 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 2 = 3 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 2 - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 3 = 4 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 2 = 3 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 2 = 3 - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 3 = 4 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1 = 2 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 2 = 3 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 3 = 4 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 2 = 3 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 2 = 3 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1 = 2 - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1 = 2 - 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 2 = 3 - 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 2 = 3 - 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 3 = 4 - 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 3 = 4 - 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 2 = 3 - 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 4 = 5 - 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 3 = 4 - 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 2 = 3 - 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 3 = 4 - 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 1 = 2 - 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 2 = 3 - 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 4 = 5 - 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 3 = 4 - 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 3 = 4 - 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 2 = 3 - 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 3 = 4 - 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 3 = 4 - 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 4 = 5 - 1
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 4 - 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 3 - 1
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 5 - 1
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 4 - 1
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 4 - 1
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 5 - 1
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 4 - 1
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 6 - 1
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 5 - 1
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 5 - 1
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 4 - 1
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 3 - 1
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 4 - 1
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 4 - 1
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 5 - 1
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 5 - 1
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 6 - 1
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 4 - 1
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 5 - 1
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 5 - 1
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 4 - 1
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 4 - 1
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 3 - 1
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 3 - 1
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 4 - 1
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 4 - 1
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 5 - 1
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 5 - 1
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 4 - 1
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 6 - 1
100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 5 - 1
101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 5 - 1
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 4 - 1
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 4 - 1
101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 3 - 1
110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 4 - 1
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 5 - 1
110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 5 - 1
110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 6 - 1
110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 4 - 1
110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 5 - 1
110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 4 - 1
111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 4 - 1
111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 5 - 1
111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 3 - 1
111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 4 - 1
0000010 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,3,4,5,8,1,6,7] => ? = 3 - 1
0000011 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,7,1,8,6] => ? = 4 - 1
0000100 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [2,3,4,7,1,5,8,6] => ? = 4 - 1
0000101 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [2,3,4,8,1,5,6,7] => ? = 3 - 1
Description
The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]
Matching statistic: St001488
(load all 4 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [[1],[]]
 => 1
1 => [1] => [[1],[]]
 => 1
00 => [2] => [[2],[]]
 => 2
01 => [1,1] => [[1,1],[]]
 => 2
10 => [1,1] => [[1,1],[]]
 => 2
11 => [2] => [[2],[]]
 => 2
000 => [3] => [[3],[]]
 => 2
001 => [2,1] => [[2,2],[1]]
 => 3
010 => [1,1,1] => [[1,1,1],[]]
 => 2
011 => [1,2] => [[2,1],[]]
 => 3
100 => [1,2] => [[2,1],[]]
 => 3
101 => [1,1,1] => [[1,1,1],[]]
 => 2
110 => [2,1] => [[2,2],[1]]
 => 3
111 => [3] => [[3],[]]
 => 2
0000 => [4] => [[4],[]]
 => 2
0001 => [3,1] => [[3,3],[2]]
 => 3
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => 3
0011 => [2,2] => [[3,2],[1]]
 => 4
0100 => [1,1,2] => [[2,1,1],[]]
 => 3
0101 => [1,1,1,1] => [[1,1,1,1],[]]
 => 2
0110 => [1,2,1] => [[2,2,1],[1]]
 => 4
0111 => [1,3] => [[3,1],[]]
 => 3
1000 => [1,3] => [[3,1],[]]
 => 3
1001 => [1,2,1] => [[2,2,1],[1]]
 => 4
1010 => [1,1,1,1] => [[1,1,1,1],[]]
 => 2
1011 => [1,1,2] => [[2,1,1],[]]
 => 3
1100 => [2,2] => [[3,2],[1]]
 => 4
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => 3
1110 => [3,1] => [[3,3],[2]]
 => 3
1111 => [4] => [[4],[]]
 => 2
00000 => [5] => [[5],[]]
 => 2
00001 => [4,1] => [[4,4],[3]]
 => 3
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => 3
00011 => [3,2] => [[4,3],[2]]
 => 4
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => 4
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => 3
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => 5
00111 => [2,3] => [[4,2],[1]]
 => 4
01000 => [1,1,3] => [[3,1,1],[]]
 => 3
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => 2
01011 => [1,1,1,2] => [[2,1,1,1],[]]
 => 3
01100 => [1,2,2] => [[3,2,1],[1]]
 => 5
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 4
01110 => [1,3,1] => [[3,3,1],[2]]
 => 4
01111 => [1,4] => [[4,1],[]]
 => 3
10000 => [1,4] => [[4,1],[]]
 => 3
10001 => [1,3,1] => [[3,3,1],[2]]
 => 4
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 4
10011 => [1,2,2] => [[3,2,1],[1]]
 => 5
000000 => [6] => [[6],[]]
 => ? = 2
000001 => [5,1] => [[5,5],[4]]
 => ? = 3
000010 => [4,1,1] => [[4,4,4],[3,3]]
 => ? = 3
000011 => [4,2] => [[5,4],[3]]
 => ? = 4
000100 => [3,1,2] => [[4,3,3],[2,2]]
 => ? = 4
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ? = 3
000110 => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
000111 => [3,3] => [[5,3],[2]]
 => ? = 4
001000 => [2,1,3] => [[4,2,2],[1,1]]
 => ? = 4
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ? = 3
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ? = 4
001100 => [2,2,2] => [[4,3,2],[2,1]]
 => ? = 6
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ? = 5
001110 => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
001111 => [2,4] => [[5,2],[1]]
 => ? = 4
010000 => [1,1,4] => [[4,1,1],[]]
 => ? = 3
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ? = 4
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ? = 5
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ? = 3
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ? = 2
010110 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ? = 4
010111 => [1,1,1,3] => [[3,1,1,1],[]]
 => ? = 3
011000 => [1,2,3] => [[4,2,1],[1]]
 => ? = 5
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ? = 4
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ? = 5
011100 => [1,3,2] => [[4,3,1],[2]]
 => ? = 5
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ? = 4
011110 => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
011111 => [1,5] => [[5,1],[]]
 => ? = 3
100000 => [1,5] => [[5,1],[]]
 => ? = 3
100001 => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ? = 4
100011 => [1,3,2] => [[4,3,1],[2]]
 => ? = 5
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ? = 5
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ? = 4
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
100111 => [1,2,3] => [[4,2,1],[1]]
 => ? = 5
101000 => [1,1,1,3] => [[3,1,1,1],[]]
 => ? = 3
101001 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ? = 4
101010 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ? = 2
101011 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ? = 3
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ? = 5
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ? = 4
101110 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
101111 => [1,1,4] => [[4,1,1],[]]
 => ? = 3
110000 => [2,4] => [[5,2],[1]]
 => ? = 4
110001 => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.
Matching statistic: St000831
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
 => [2,1] => 1
1 => [1] => [1,0]
 => [2,1] => 1
00 => [2] => [1,1,0,0]
 => [2,3,1] => 2
01 => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
10 => [1,1] => [1,0,1,0]
 => [3,1,2] => 2
11 => [2] => [1,1,0,0]
 => [2,3,1] => 2
000 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 2
001 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
010 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 2
011 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
100 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 3
101 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 2
110 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 3
111 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 2
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 4
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 4
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 4
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 4
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 2
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 2
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 3
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 3
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 4
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 4
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 5
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 4
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 3
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 4
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 3
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 5
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 4
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 5
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ? = 2
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 3
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 3
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ? = 4
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ? = 4
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 3
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 5
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ? = 4
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ? = 4
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ? = 5
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 3
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ? = 4
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ? = 6
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 5
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 5
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 4
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 3
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 4
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 4
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 5
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 3
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 2
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 4
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 3
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 6
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 4
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 5
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 5
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 4
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 4
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 3
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 3
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 4
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 4
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ? = 5
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 5
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 4
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 6
100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 5
101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 3
101001 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 4
101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 2
101011 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 3
101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 5
101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 4
101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 4
101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 3
110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ? = 4
110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 5
Description
The number of indices that are either descents or recoils. This is, for a permutation $\pi$ of length $n$, this statistics counts the set $$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$