Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001524
(load all 6 compositions to match this statistic)
Mapping
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 1
01 => 0
10 => 0
11 => 1
000 => 1
001 => 0
010 => 1
011 => 0
100 => 0
101 => 1
110 => 0
111 => 1
0000 => 2
0001 => 1
0010 => 1
0011 => 0
0100 => 1
0101 => 0
0110 => 2
0111 => 1
1000 => 1
1001 => 2
1010 => 0
1011 => 1
1100 => 0
1101 => 1
1110 => 1
1111 => 2
00000 => 2
00001 => 1
00010 => 1
00011 => 0
00100 => 2
00101 => 1
00110 => 1
00111 => 0
01000 => 1
01001 => 0
01010 => 2
01011 => 1
01100 => 1
01101 => 0
01110 => 2
01111 => 1
10000 => 1
10001 => 2
10010 => 0
10011 => 1
Description
The degree of symmetry of a binary word. For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Matching statistic: St001092
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => []
 => 0
1 => [1,1] => [1,0,1,0]
 => [1]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => []
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 2
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,4]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 0
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 0
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St001153
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => {{1}}
 => 0
1 => [1] => [1,0]
 => {{1}}
 => 0
00 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
01 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
10 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
11 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
000 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
001 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
010 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
011 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
100 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
110 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
111 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2
Description
The number of blocks with even minimum in a set partition.
Matching statistic: St001151
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 2 = 1 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 2 = 1 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2 = 1 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1 = 0 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 2 = 1 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 3 = 2 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 3 = 2 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 3 = 2 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2,3,4},{5},{6}}
 => 2 = 1 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 1 = 0 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 1 = 0 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6}}
 => 2 = 1 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 3 = 2 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 3 = 2 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 3 = 2 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => 2 = 1 + 1
Description
The number of blocks with odd minimum. See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St001114
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => [1] => ? ∊ {0,0}
1 => [1] => [1,0]
 => [1] => ? ∊ {0,0}
00 => [2] => [1,1,0,0]
 => [1,2] => 0
01 => [1,1] => [1,0,1,0]
 => [2,1] => 1
10 => [1,1] => [1,0,1,0]
 => [2,1] => 1
11 => [2] => [1,1,0,0]
 => [1,2] => 0
000 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
001 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 1
011 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 0
100 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 0
101 => [1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 1
110 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 1
111 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 0
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 0
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 2
Description
The number of odd descents of a permutation.
Matching statistic: St001115
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,1] => 0
1 => [1,1] => [1,0,1,0]
 => [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 2
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 2
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,1,6,5,4] => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,6] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 0
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 0
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 0
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,2,1,4,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,2,1,4,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,5,4,3,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,4,3,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,4,3,2,6,7] => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
Description
The number of even descents of a permutation.
Matching statistic: St000259
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 75%
Values
0 => [1] => ([],1)
 => ([],1)
 => 0
1 => [1] => ([],1)
 => ([],1)
 => 0
00 => [2] => ([],2)
 => ([],1)
 => 0
01 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
10 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
11 => [2] => ([],2)
 => ([],1)
 => 0
000 => [3] => ([],3)
 => ([],1)
 => 0
001 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
011 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,0}
100 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {0,0}
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
111 => [3] => ([],3)
 => ([],1)
 => 0
0000 => [4] => ([],4)
 => ([],1)
 => 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
0011 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,1,1,2,2}
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,1,1,2,2}
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
0111 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,1,1,2,2}
1000 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,1,1,2,2}
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,1,1,2,2}
1100 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {0,0,1,1,2,2}
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
1111 => [4] => ([],4)
 => ([],1)
 => 0
00000 => [5] => ([],5)
 => ([],1)
 => 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
00111 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
01111 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
10000 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
11000 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1}
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
11111 => [5] => ([],5)
 => ([],1)
 => 0
000000 => [6] => ([],6)
 => ([],1)
 => 0
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
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)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
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)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
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)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
001111 => [2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
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)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
011111 => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
100000 => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
101000 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
101011 => [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)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
110000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
110011 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
110100 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
110111 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
111000 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000620
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000620: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [[1],[]]
 => []
 => ? ∊ {0,0}
1 => [1] => [[1],[]]
 => []
 => ? ∊ {0,0}
00 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
001 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1}
010 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
011 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
100 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
101 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
110 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1}
111 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
0000 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
0001 => [3,1] => [[3,3],[2]]
 => [2]
 => 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
0011 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
0111 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
1000 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
1100 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
1110 => [3,1] => [[3,3],[2]]
 => [2]
 => 0
1111 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2}
00000 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
00001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
00011 => [3,2] => [[4,3],[2]]
 => [2]
 => 0
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
00111 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
01000 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
01100 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
01110 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
01111 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
10000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
10001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
10011 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
10111 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
11000 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
11011 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
11100 => [3,2] => [[4,3],[2]]
 => [2]
 => 0
11101 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
11110 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
11111 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2}
000000 => [6] => [[6],[]]
 => []
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
000001 => [5,1] => [[5,5],[4]]
 => [4]
 => 0
000010 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
000011 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
000100 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
000110 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 3
000111 => [3,3] => [[5,3],[2]]
 => [2]
 => 0
001000 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 2
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 3
001110 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
001111 => [2,4] => [[5,2],[1]]
 => [1]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010000 => [1,1,4] => [[4,1,1],[]]
 => []
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010110 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
010111 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
011100 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1
011110 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
100001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1
100011 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
101110 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].
Matching statistic: St000698
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [[1],[]]
 => []
 => ? ∊ {0,0}
1 => [1] => [[1],[]]
 => []
 => ? ∊ {0,0}
00 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
 => []
 => ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
001 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1}
010 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
011 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
100 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
101 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
110 => [2,1] => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1}
111 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1}
0000 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
0001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
0011 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
0111 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
1000 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
1100 => [2,2] => [[3,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
1110 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
1111 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2}
00000 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
00001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
00011 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
00111 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
01000 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
01100 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
01110 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
01111 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
10000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
10001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
10011 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
10111 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
11000 => [2,3] => [[4,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 0
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
11011 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
11100 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
11101 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
11110 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
11111 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2}
000000 => [6] => [[6],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
000001 => [5,1] => [[5,5],[4]]
 => [4]
 => 2
000010 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
000011 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
000100 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
000110 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 2
000111 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
001000 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 2
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 2
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 0
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 2
001110 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
001111 => [2,4] => [[5,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
010000 => [1,1,4] => [[4,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
010110 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
010111 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 0
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
011100 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
011110 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
100001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
100011 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
101110 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$. This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
 => ([],1)
 => ? ∊ {0,0}
1 => [1,1] => [[1,1],[]]
 => ([],1)
 => ? ∊ {0,0}
00 => [3] => [[3],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
01 => [2,1] => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1}
10 => [1,2] => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
11 => [1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1}
000 => [4] => [[4],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1}
001 => [3,1] => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1}
010 => [2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1}
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1}
100 => [1,3] => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1}
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1}
110 => [1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1}
0000 => [5] => [[5],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
0001 => [4,1] => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
0010 => [3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
0100 => [2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1000 => [1,4] => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1100 => [1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2}
00000 => [6] => [[6],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
00001 => [5,1] => [[5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
00010 => [4,2] => [[5,4],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
00100 => [3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
01000 => [2,4] => [[5,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
10000 => [1,5] => [[5,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
10001 => [1,4,1] => [[4,4,1],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
10010 => [1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
10100 => [1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
11000 => [1,1,4] => [[4,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
11100 => [1,1,1,3] => [[3,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2}
000000 => [7] => [[7],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
000001 => [6,1] => [[6,6],[5]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
000100 => [4,3] => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 1
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => 1
001000 => [3,4] => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
001010 => [3,2,2] => [[5,4,3],[3,2]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => ([(0,2),(2,1)],3)
 => 1
001100 => [3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
010001 => [2,4,1] => [[5,5,2],[4,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
010010 => [2,3,2] => [[5,4,2],[3,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
010100 => [2,2,3] => [[5,3,2],[2,1]]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 1
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => ([(0,2),(2,1)],3)
 => 1
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 2
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
100010 => [1,4,2] => [[5,4,1],[3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
100100 => [1,3,3] => [[5,3,1],[2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 1
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
101000 => [1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 1
101010 => [1,2,2,2] => [[4,3,2,1],[2,1]]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 1
101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
110010 => [1,1,3,2] => [[4,3,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 1
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,2),(2,1)],3)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000455The second largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000544The cop number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001399The distinguishing number of a poset. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000454The largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice.