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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000899
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000899: 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] => 1
01 => [1,1] => 2
10 => [1,1] => 2
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 1
010 => [1,1,1] => 3
011 => [1,2] => 1
100 => [1,2] => 1
101 => [1,1,1] => 3
110 => [2,1] => 1
111 => [3] => 1
0000 => [4] => 1
0001 => [3,1] => 1
0010 => [2,1,1] => 2
0011 => [2,2] => 2
0100 => [1,1,2] => 2
0101 => [1,1,1,1] => 4
0110 => [1,2,1] => 1
0111 => [1,3] => 1
1000 => [1,3] => 1
1001 => [1,2,1] => 1
1010 => [1,1,1,1] => 4
1011 => [1,1,2] => 2
1100 => [2,2] => 2
1101 => [2,1,1] => 2
1110 => [3,1] => 1
1111 => [4] => 1
00000 => [5] => 1
00001 => [4,1] => 1
00010 => [3,1,1] => 2
00011 => [3,2] => 1
00100 => [2,1,2] => 1
00101 => [2,1,1,1] => 3
00110 => [2,2,1] => 2
00111 => [2,3] => 1
01000 => [1,1,3] => 2
01001 => [1,1,2,1] => 2
01010 => [1,1,1,1,1] => 5
01011 => [1,1,1,2] => 3
01100 => [1,2,2] => 2
01101 => [1,2,1,1] => 2
01110 => [1,3,1] => 1
01111 => [1,4] => 1
10000 => [1,4] => 1
10001 => [1,3,1] => 1
10010 => [1,2,1,1] => 2
10011 => [1,2,2] => 2
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000381
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 1
1 => [1] => [1] => 1
00 => [2] => [1] => 1
01 => [1,1] => [2] => 2
10 => [1,1] => [2] => 2
11 => [2] => [1] => 1
000 => [3] => [1] => 1
001 => [2,1] => [1,1] => 1
010 => [1,1,1] => [3] => 3
011 => [1,2] => [1,1] => 1
100 => [1,2] => [1,1] => 1
101 => [1,1,1] => [3] => 3
110 => [2,1] => [1,1] => 1
111 => [3] => [1] => 1
0000 => [4] => [1] => 1
0001 => [3,1] => [1,1] => 1
0010 => [2,1,1] => [1,2] => 2
0011 => [2,2] => [2] => 2
0100 => [1,1,2] => [2,1] => 2
0101 => [1,1,1,1] => [4] => 4
0110 => [1,2,1] => [1,1,1] => 1
0111 => [1,3] => [1,1] => 1
1000 => [1,3] => [1,1] => 1
1001 => [1,2,1] => [1,1,1] => 1
1010 => [1,1,1,1] => [4] => 4
1011 => [1,1,2] => [2,1] => 2
1100 => [2,2] => [2] => 2
1101 => [2,1,1] => [1,2] => 2
1110 => [3,1] => [1,1] => 1
1111 => [4] => [1] => 1
00000 => [5] => [1] => 1
00001 => [4,1] => [1,1] => 1
00010 => [3,1,1] => [1,2] => 2
00011 => [3,2] => [1,1] => 1
00100 => [2,1,2] => [1,1,1] => 1
00101 => [2,1,1,1] => [1,3] => 3
00110 => [2,2,1] => [2,1] => 2
00111 => [2,3] => [1,1] => 1
01000 => [1,1,3] => [2,1] => 2
01001 => [1,1,2,1] => [2,1,1] => 2
01010 => [1,1,1,1,1] => [5] => 5
01011 => [1,1,1,2] => [3,1] => 3
01100 => [1,2,2] => [1,2] => 2
01101 => [1,2,1,1] => [1,1,2] => 2
01110 => [1,3,1] => [1,1,1] => 1
01111 => [1,4] => [1,1] => 1
10000 => [1,4] => [1,1] => 1
10001 => [1,3,1] => [1,1,1] => 1
10010 => [1,2,1,1] => [1,1,2] => 2
10011 => [1,2,2] => [1,2] => 2
Description
The largest part of an integer composition.
Matching statistic: St000147
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => [1]
=> 1
1 => [1] => [1] => [1]
=> 1
00 => [2] => [1] => [1]
=> 1
01 => [1,1] => [2] => [2]
=> 2
10 => [1,1] => [2] => [2]
=> 2
11 => [2] => [1] => [1]
=> 1
000 => [3] => [1] => [1]
=> 1
001 => [2,1] => [1,1] => [1,1]
=> 1
010 => [1,1,1] => [3] => [3]
=> 3
011 => [1,2] => [1,1] => [1,1]
=> 1
100 => [1,2] => [1,1] => [1,1]
=> 1
101 => [1,1,1] => [3] => [3]
=> 3
110 => [2,1] => [1,1] => [1,1]
=> 1
111 => [3] => [1] => [1]
=> 1
0000 => [4] => [1] => [1]
=> 1
0001 => [3,1] => [1,1] => [1,1]
=> 1
0010 => [2,1,1] => [1,2] => [2,1]
=> 2
0011 => [2,2] => [2] => [2]
=> 2
0100 => [1,1,2] => [2,1] => [2,1]
=> 2
0101 => [1,1,1,1] => [4] => [4]
=> 4
0110 => [1,2,1] => [1,1,1] => [1,1,1]
=> 1
0111 => [1,3] => [1,1] => [1,1]
=> 1
1000 => [1,3] => [1,1] => [1,1]
=> 1
1001 => [1,2,1] => [1,1,1] => [1,1,1]
=> 1
1010 => [1,1,1,1] => [4] => [4]
=> 4
1011 => [1,1,2] => [2,1] => [2,1]
=> 2
1100 => [2,2] => [2] => [2]
=> 2
1101 => [2,1,1] => [1,2] => [2,1]
=> 2
1110 => [3,1] => [1,1] => [1,1]
=> 1
1111 => [4] => [1] => [1]
=> 1
00000 => [5] => [1] => [1]
=> 1
00001 => [4,1] => [1,1] => [1,1]
=> 1
00010 => [3,1,1] => [1,2] => [2,1]
=> 2
00011 => [3,2] => [1,1] => [1,1]
=> 1
00100 => [2,1,2] => [1,1,1] => [1,1,1]
=> 1
00101 => [2,1,1,1] => [1,3] => [3,1]
=> 3
00110 => [2,2,1] => [2,1] => [2,1]
=> 2
00111 => [2,3] => [1,1] => [1,1]
=> 1
01000 => [1,1,3] => [2,1] => [2,1]
=> 2
01001 => [1,1,2,1] => [2,1,1] => [2,1,1]
=> 2
01010 => [1,1,1,1,1] => [5] => [5]
=> 5
01011 => [1,1,1,2] => [3,1] => [3,1]
=> 3
01100 => [1,2,2] => [1,2] => [2,1]
=> 2
01101 => [1,2,1,1] => [1,1,2] => [2,1,1]
=> 2
01110 => [1,3,1] => [1,1,1] => [1,1,1]
=> 1
01111 => [1,4] => [1,1] => [1,1]
=> 1
10000 => [1,4] => [1,1] => [1,1]
=> 1
10001 => [1,3,1] => [1,1,1] => [1,1,1]
=> 1
10010 => [1,2,1,1] => [1,1,2] => [2,1,1]
=> 2
10011 => [1,2,2] => [1,2] => [2,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000013
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => [1,0]
=> 1
1 => [1] => [1] => [1,0]
=> 1
00 => [2] => [1] => [1,0]
=> 1
01 => [1,1] => [2] => [1,1,0,0]
=> 2
10 => [1,1] => [2] => [1,1,0,0]
=> 2
11 => [2] => [1] => [1,0]
=> 1
000 => [3] => [1] => [1,0]
=> 1
001 => [2,1] => [1,1] => [1,0,1,0]
=> 1
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
011 => [1,2] => [1,1] => [1,0,1,0]
=> 1
100 => [1,2] => [1,1] => [1,0,1,0]
=> 1
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
110 => [2,1] => [1,1] => [1,0,1,0]
=> 1
111 => [3] => [1] => [1,0]
=> 1
0000 => [4] => [1] => [1,0]
=> 1
0001 => [3,1] => [1,1] => [1,0,1,0]
=> 1
0010 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
0011 => [2,2] => [2] => [1,1,0,0]
=> 2
0100 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
0110 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
0111 => [1,3] => [1,1] => [1,0,1,0]
=> 1
1000 => [1,3] => [1,1] => [1,0,1,0]
=> 1
1001 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
1010 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
1011 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
1100 => [2,2] => [2] => [1,1,0,0]
=> 2
1101 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
1110 => [3,1] => [1,1] => [1,0,1,0]
=> 1
1111 => [4] => [1] => [1,0]
=> 1
00000 => [5] => [1] => [1,0]
=> 1
00001 => [4,1] => [1,1] => [1,0,1,0]
=> 1
00010 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
00011 => [3,2] => [1,1] => [1,0,1,0]
=> 1
00100 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1
00101 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
00110 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
00111 => [2,3] => [1,1] => [1,0,1,0]
=> 1
01000 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
01001 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
01011 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
01100 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
01101 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
01110 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
01111 => [1,4] => [1,1] => [1,0,1,0]
=> 1
10000 => [1,4] => [1,1] => [1,0,1,0]
=> 1
10001 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
10010 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
10011 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
010100101 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
010101101 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
010110101 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
101001010 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
101010010 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
101011010 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000444
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%
Values
0 => [1] => [1] => [1,0]
=> ? = 1
1 => [1] => [1] => [1,0]
=> ? = 1
00 => [2] => [1] => [1,0]
=> ? = 1
01 => [1,1] => [2] => [1,1,0,0]
=> 2
10 => [1,1] => [2] => [1,1,0,0]
=> 2
11 => [2] => [1] => [1,0]
=> ? = 1
000 => [3] => [1] => [1,0]
=> ? = 1
001 => [2,1] => [1,1] => [1,0,1,0]
=> 1
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
011 => [1,2] => [1,1] => [1,0,1,0]
=> 1
100 => [1,2] => [1,1] => [1,0,1,0]
=> 1
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
110 => [2,1] => [1,1] => [1,0,1,0]
=> 1
111 => [3] => [1] => [1,0]
=> ? = 1
0000 => [4] => [1] => [1,0]
=> ? = 1
0001 => [3,1] => [1,1] => [1,0,1,0]
=> 1
0010 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
0011 => [2,2] => [2] => [1,1,0,0]
=> 2
0100 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
0110 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
0111 => [1,3] => [1,1] => [1,0,1,0]
=> 1
1000 => [1,3] => [1,1] => [1,0,1,0]
=> 1
1001 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
1010 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
1011 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
1100 => [2,2] => [2] => [1,1,0,0]
=> 2
1101 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
1110 => [3,1] => [1,1] => [1,0,1,0]
=> 1
1111 => [4] => [1] => [1,0]
=> ? = 1
00000 => [5] => [1] => [1,0]
=> ? = 1
00001 => [4,1] => [1,1] => [1,0,1,0]
=> 1
00010 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
00011 => [3,2] => [1,1] => [1,0,1,0]
=> 1
00100 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1
00101 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
00110 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
00111 => [2,3] => [1,1] => [1,0,1,0]
=> 1
01000 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
01001 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
01011 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
01100 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
01101 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
01110 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
01111 => [1,4] => [1,1] => [1,0,1,0]
=> 1
10000 => [1,4] => [1,1] => [1,0,1,0]
=> 1
10001 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
10010 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
10011 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
10100 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
10101 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
10110 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
10111 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
11000 => [2,3] => [1,1] => [1,0,1,0]
=> 1
11001 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
11010 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
11011 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1
11100 => [3,2] => [1,1] => [1,0,1,0]
=> 1
11111 => [5] => [1] => [1,0]
=> ? = 1
000000 => [6] => [1] => [1,0]
=> ? = 1
111111 => [6] => [1] => [1,0]
=> ? = 1
0000000 => [7] => [1] => [1,0]
=> ? = 1
1111111 => [7] => [1] => [1,0]
=> ? = 1
00000000 => [8] => [1] => [1,0]
=> ? = 1
01010101 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
10101010 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
11111111 => [8] => [1] => [1,0]
=> ? = 1
000000000 => [9] => [1] => [1,0]
=> ? = 1
010100101 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
010101001 => [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
010101010 => [1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
010101011 => [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
010101101 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
010110101 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
101001010 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
101010010 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
101010100 => [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
101010101 => [1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
101010110 => [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
101011010 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
111111111 => [9] => [1] => [1,0]
=> ? = 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 97%●distinct values known / distinct values provided: 78%
Values
0 => [1] => [1] => [1,0]
=> ? = 1 - 1
1 => [1] => [1] => [1,0]
=> ? = 1 - 1
00 => [2] => [1] => [1,0]
=> ? = 1 - 1
01 => [1,1] => [2] => [1,1,0,0]
=> 1 = 2 - 1
10 => [1,1] => [2] => [1,1,0,0]
=> 1 = 2 - 1
11 => [2] => [1] => [1,0]
=> ? = 1 - 1
000 => [3] => [1] => [1,0]
=> ? = 1 - 1
001 => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
011 => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
100 => [1,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
110 => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
111 => [3] => [1] => [1,0]
=> ? = 1 - 1
0000 => [4] => [1] => [1,0]
=> ? = 1 - 1
0001 => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
0010 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
0011 => [2,2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
0100 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
0110 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
0111 => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
1000 => [1,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
1001 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
1010 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
1011 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
1100 => [2,2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
1101 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
1110 => [3,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
1111 => [4] => [1] => [1,0]
=> ? = 1 - 1
00000 => [5] => [1] => [1,0]
=> ? = 1 - 1
00001 => [4,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
00010 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
00011 => [3,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
00100 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
00101 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
00110 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
00111 => [2,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
01000 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
01001 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
01011 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
01100 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
01101 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
01110 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
01111 => [1,4] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
10000 => [1,4] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
10001 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
10010 => [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
10011 => [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
10100 => [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
10101 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
10110 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
10111 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
11000 => [2,3] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
11001 => [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
11010 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
11011 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
11100 => [3,2] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
11111 => [5] => [1] => [1,0]
=> ? = 1 - 1
000000 => [6] => [1] => [1,0]
=> ? = 1 - 1
111111 => [6] => [1] => [1,0]
=> ? = 1 - 1
0000000 => [7] => [1] => [1,0]
=> ? = 1 - 1
1111111 => [7] => [1] => [1,0]
=> ? = 1 - 1
00000000 => [8] => [1] => [1,0]
=> ? = 1 - 1
01010101 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
10101010 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
11111111 => [8] => [1] => [1,0]
=> ? = 1 - 1
000000000 => [9] => [1] => [1,0]
=> ? = 1 - 1
010100101 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
010101001 => [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
010101010 => [1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
010101011 => [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
010101101 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
010110101 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
101001010 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
101010010 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
101010100 => [1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
101010101 => [1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
101010110 => [1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
101011010 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
111111111 => [9] => [1] => [1,0]
=> ? = 1 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001235
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%
Values
0 => [1] => [1] => [1] => 1
1 => [1] => [1] => [1] => 1
00 => [2] => [1] => [1] => 1
01 => [1,1] => [2] => [1,1] => 2
10 => [1,1] => [2] => [1,1] => 2
11 => [2] => [1] => [1] => 1
000 => [3] => [1] => [1] => 1
001 => [2,1] => [1,1] => [2] => 1
010 => [1,1,1] => [3] => [1,1,1] => 3
011 => [1,2] => [1,1] => [2] => 1
100 => [1,2] => [1,1] => [2] => 1
101 => [1,1,1] => [3] => [1,1,1] => 3
110 => [2,1] => [1,1] => [2] => 1
111 => [3] => [1] => [1] => 1
0000 => [4] => [1] => [1] => 1
0001 => [3,1] => [1,1] => [2] => 1
0010 => [2,1,1] => [1,2] => [1,2] => 2
0011 => [2,2] => [2] => [1,1] => 2
0100 => [1,1,2] => [2,1] => [2,1] => 2
0101 => [1,1,1,1] => [4] => [1,1,1,1] => 4
0110 => [1,2,1] => [1,1,1] => [3] => 1
0111 => [1,3] => [1,1] => [2] => 1
1000 => [1,3] => [1,1] => [2] => 1
1001 => [1,2,1] => [1,1,1] => [3] => 1
1010 => [1,1,1,1] => [4] => [1,1,1,1] => 4
1011 => [1,1,2] => [2,1] => [2,1] => 2
1100 => [2,2] => [2] => [1,1] => 2
1101 => [2,1,1] => [1,2] => [1,2] => 2
1110 => [3,1] => [1,1] => [2] => 1
1111 => [4] => [1] => [1] => 1
00000 => [5] => [1] => [1] => 1
00001 => [4,1] => [1,1] => [2] => 1
00010 => [3,1,1] => [1,2] => [1,2] => 2
00011 => [3,2] => [1,1] => [2] => 1
00100 => [2,1,2] => [1,1,1] => [3] => 1
00101 => [2,1,1,1] => [1,3] => [1,1,2] => 3
00110 => [2,2,1] => [2,1] => [2,1] => 2
00111 => [2,3] => [1,1] => [2] => 1
01000 => [1,1,3] => [2,1] => [2,1] => 2
01001 => [1,1,2,1] => [2,1,1] => [3,1] => 2
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1] => 5
01011 => [1,1,1,2] => [3,1] => [2,1,1] => 3
01100 => [1,2,2] => [1,2] => [1,2] => 2
01101 => [1,2,1,1] => [1,1,2] => [1,3] => 2
01110 => [1,3,1] => [1,1,1] => [3] => 1
01111 => [1,4] => [1,1] => [2] => 1
10000 => [1,4] => [1,1] => [2] => 1
10001 => [1,3,1] => [1,1,1] => [3] => 1
10010 => [1,2,1,1] => [1,1,2] => [1,3] => 2
10011 => [1,2,2] => [1,2] => [1,2] => 2
0101010 => [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ? = 7
1010101 => [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ? = 7
00101010 => [2,1,1,1,1,1,1] => [1,6] => [1,1,1,1,1,2] => ? = 6
01001010 => [1,1,2,1,1,1,1] => [2,1,4] => [1,1,1,3,1] => ? = 4
01010010 => [1,1,1,1,2,1,1] => [4,1,2] => [1,3,1,1,1] => ? = 4
01010100 => [1,1,1,1,1,1,2] => [6,1] => [2,1,1,1,1,1] => ? = 6
01010101 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1] => ? = 8
01010110 => [1,1,1,1,1,2,1] => [5,1,1] => [3,1,1,1,1] => ? = 5
01011010 => [1,1,1,2,1,1,1] => [3,1,3] => [1,1,3,1,1] => ? = 3
01101010 => [1,2,1,1,1,1,1] => [1,1,5] => [1,1,1,1,3] => ? = 5
10010101 => [1,2,1,1,1,1,1] => [1,1,5] => [1,1,1,1,3] => ? = 5
10100101 => [1,1,1,2,1,1,1] => [3,1,3] => [1,1,3,1,1] => ? = 3
10101001 => [1,1,1,1,1,2,1] => [5,1,1] => [3,1,1,1,1] => ? = 5
10101010 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1] => ? = 8
10101011 => [1,1,1,1,1,1,2] => [6,1] => [2,1,1,1,1,1] => ? = 6
10101101 => [1,1,1,1,2,1,1] => [4,1,2] => [1,3,1,1,1] => ? = 4
10110101 => [1,1,2,1,1,1,1] => [2,1,4] => [1,1,1,3,1] => ? = 4
11010101 => [2,1,1,1,1,1,1] => [1,6] => [1,1,1,1,1,2] => ? = 6
000101010 => [3,1,1,1,1,1,1] => [1,6] => [1,1,1,1,1,2] => ? = 6
001001010 => [2,1,2,1,1,1,1] => [1,1,1,4] => [1,1,1,4] => ? = 4
001010010 => [2,1,1,1,2,1,1] => [1,3,1,2] => [1,3,1,2] => ? = 3
001010100 => [2,1,1,1,1,1,2] => [1,5,1] => [2,1,1,1,2] => ? = 5
001010101 => [2,1,1,1,1,1,1,1] => [1,7] => [1,1,1,1,1,1,2] => ? = 7
001010110 => [2,1,1,1,1,2,1] => [1,4,1,1] => [3,1,1,2] => ? = 4
001011010 => [2,1,1,2,1,1,1] => [1,2,1,3] => [1,1,3,2] => ? = 3
001101010 => [2,2,1,1,1,1,1] => [2,5] => [1,1,1,1,2,1] => ? = 5
010001010 => [1,1,3,1,1,1,1] => [2,1,4] => [1,1,1,3,1] => ? = 4
010010010 => [1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,5,1] => ? = 2
010010100 => [1,1,2,1,1,1,2] => [2,1,3,1] => [2,1,3,1] => ? = 3
010010101 => [1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,1,1,3,1] => ? = 5
010010110 => [1,1,2,1,1,2,1] => [2,1,2,1,1] => [3,3,1] => ? = 2
010011010 => [1,1,2,2,1,1,1] => [2,2,3] => [1,1,2,2,1] => ? = 3
010100010 => [1,1,1,1,3,1,1] => [4,1,2] => [1,3,1,1,1] => ? = 4
010100100 => [1,1,1,1,2,1,2] => [4,1,1,1] => [4,1,1,1] => ? = 4
010100101 => [1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,3,1,1,1] => ? = 4
010100110 => [1,1,1,1,2,2,1] => [4,2,1] => [2,2,1,1,1] => ? = 4
010101000 => [1,1,1,1,1,1,3] => [6,1] => [2,1,1,1,1,1] => ? = 6
010101001 => [1,1,1,1,1,1,2,1] => [6,1,1] => [3,1,1,1,1,1] => ? = 6
010101010 => [1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1] => ? = 9
010101011 => [1,1,1,1,1,1,1,2] => [7,1] => [2,1,1,1,1,1,1] => ? = 7
010101100 => [1,1,1,1,1,2,2] => [5,2] => [1,2,1,1,1,1] => ? = 5
010101101 => [1,1,1,1,1,2,1,1] => [5,1,2] => [1,3,1,1,1,1] => ? = 5
010101110 => [1,1,1,1,1,3,1] => [5,1,1] => [3,1,1,1,1] => ? = 5
010110010 => [1,1,1,2,2,1,1] => [3,2,2] => [1,2,2,1,1] => ? = 3
010110100 => [1,1,1,2,1,1,2] => [3,1,2,1] => [2,3,1,1] => ? = 3
010110101 => [1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,3,1,1] => ? = 4
010110110 => [1,1,1,2,1,2,1] => [3,1,1,1,1] => [5,1,1] => ? = 3
010111010 => [1,1,1,3,1,1,1] => [3,1,3] => [1,1,3,1,1] => ? = 3
011001010 => [1,2,2,1,1,1,1] => [1,2,4] => [1,1,1,2,2] => ? = 4
011010010 => [1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,3,3] => ? = 2
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001330
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 78%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 78%
Values
0 => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
1 => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
00 => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
01 => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
10 => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
11 => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
000 => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
011 => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
100 => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
111 => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
0000 => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
0111 => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
1000 => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
1111 => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
00000 => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? = 2 + 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 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)
=> ([(0,2),(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 + 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 + 1
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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
01111 => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
10000 => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 + 1
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(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 + 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? = 2 + 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
11111 => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
000000 => [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
101010 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
111111 => [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
0000000 => [7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
0101010 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 7 + 1
1010101 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 7 + 1
1111111 => [7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000475
Mp00262: Binary words —poset of factors⟶ Posets
Mp00332: Posets —Jordan block partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00332: Posets —Jordan block partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
0 => ([(0,1)],2)
=> [2]
=> 0 = 1 - 1
1 => ([(0,1)],2)
=> [2]
=> 0 = 1 - 1
00 => ([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
11 => ([(0,2),(2,1)],3)
=> [3]
=> 0 = 1 - 1
000 => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0 = 1 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,1,1]
=> 2 = 3 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0 = 1 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0 = 1 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,1,1]
=> 2 = 3 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 0 = 1 - 1
111 => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0 = 1 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ?
=> ? = 1 - 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ?
=> ? = 2 - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? = 2 - 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ?
=> ? = 2 - 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ?
=> ? = 4 - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ?
=> ? = 1 - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ?
=> ? = 1 - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ?
=> ? = 1 - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ?
=> ? = 1 - 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ?
=> ? = 4 - 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ?
=> ? = 2 - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? = 2 - 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ?
=> ? = 2 - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ?
=> ? = 1 - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 0 = 1 - 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 0 = 1 - 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1 - 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 2 - 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 1 - 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ?
=> ? = 1 - 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ?
=> ? = 3 - 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ?
=> ? = 2 - 1
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 1 - 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 2 - 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ?
=> ? = 2 - 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ?
=> ? = 5 - 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ?
=> ? = 3 - 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ?
=> ? = 2 - 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ?
=> ? = 2 - 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ?
=> ? = 1 - 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1 - 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1 - 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ?
=> ? = 1 - 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ?
=> ? = 2 - 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ?
=> ? = 2 - 1
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ?
=> ? = 3 - 1
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ?
=> ? = 5 - 1
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ?
=> ? = 2 - 1
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 2 - 1
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 1 - 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ?
=> ? = 2 - 1
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ?
=> ? = 3 - 1
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ?
=> ? = 1 - 1
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 1 - 1
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 2 - 1
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1 - 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 0 = 1 - 1
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 0 = 1 - 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ?
=> ? = 1 - 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ?
=> ? = 2 - 1
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ?
=> ? = 1 - 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ?
=> ? = 1 - 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ?
=> ? = 3 - 1
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ?
=> ? = 1 - 1
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 0 = 1 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000907
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000907: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00262: Binary words —poset of factors⟶ Posets
St000907: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
0 => 0 => ([(0,1)],2)
=> 2 = 1 + 1
1 => 1 => ([(0,1)],2)
=> 2 = 1 + 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
01 => 00 => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
10 => 11 => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
1000 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 2 + 1
1101 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
1110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1 + 1
00001 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
00010 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
00011 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
00100 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
00101 => 01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 + 1
00110 => 01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
00111 => 01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
01000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
01001 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
01011 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 + 1
01100 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
01101 => 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
01110 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
01111 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
10000 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
10001 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
10010 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
10011 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
10100 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 + 1
10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
10110 => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
10111 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
11000 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
11001 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
11010 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 + 1
11011 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
11100 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
11101 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
11110 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1 + 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1 + 1
000001 => 010100 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 1 + 1
000010 => 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 2 + 1
000011 => 010110 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 1 + 1
000100 => 010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 1 + 1
000101 => 010000 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 3 + 1
010101 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
Description
The number of maximal antichains of minimal length in a poset.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001354The number of series nodes in the modular decomposition of a graph. St000553The number of blocks of a graph. St000552The number of cut vertices of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001691The number of kings in a graph.
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