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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001151
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%
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
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 2
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 3
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 3
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3
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
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
Description
The number of blocks with odd minimum.
See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St001092
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%
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 => [1,1] => [1,0,1,0]
=> [1]
=> 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
01 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0 = 1 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 1 = 2 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0 = 1 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0 = 1 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 0 = 1 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1 = 2 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1 = 2 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 2 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0 = 1 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 2 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 3 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 2 = 3 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0 = 1 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 2 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0 = 1 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1 = 2 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1 = 2 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2 = 3 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1 = 2 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 2 = 3 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0 = 1 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0 = 1 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1 = 2 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 0 = 1 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 0 = 1 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 1 = 2 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 1 = 2 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> 2 = 3 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 1 = 2 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> 2 = 3 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 2 = 3 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 0 = 1 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 0 = 1 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 1 = 2 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1 = 2 - 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
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