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Matching statistic: St000147
Mp00097: Binary words —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%
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 = 2 - 1
1 => [1] => [1]
=> 1 = 2 - 1
00 => [2] => [2]
=> 2 = 3 - 1
01 => [1,1] => [1,1]
=> 1 = 2 - 1
10 => [1,1] => [1,1]
=> 1 = 2 - 1
11 => [2] => [2]
=> 2 = 3 - 1
000 => [3] => [3]
=> 3 = 4 - 1
001 => [2,1] => [2,1]
=> 2 = 3 - 1
010 => [1,1,1] => [1,1,1]
=> 1 = 2 - 1
011 => [1,2] => [2,1]
=> 2 = 3 - 1
100 => [1,2] => [2,1]
=> 2 = 3 - 1
101 => [1,1,1] => [1,1,1]
=> 1 = 2 - 1
110 => [2,1] => [2,1]
=> 2 = 3 - 1
111 => [3] => [3]
=> 3 = 4 - 1
0000 => [4] => [4]
=> 4 = 5 - 1
0001 => [3,1] => [3,1]
=> 3 = 4 - 1
0010 => [2,1,1] => [2,1,1]
=> 2 = 3 - 1
0011 => [2,2] => [2,2]
=> 2 = 3 - 1
0100 => [1,1,2] => [2,1,1]
=> 2 = 3 - 1
0101 => [1,1,1,1] => [1,1,1,1]
=> 1 = 2 - 1
0110 => [1,2,1] => [2,1,1]
=> 2 = 3 - 1
0111 => [1,3] => [3,1]
=> 3 = 4 - 1
1000 => [1,3] => [3,1]
=> 3 = 4 - 1
1001 => [1,2,1] => [2,1,1]
=> 2 = 3 - 1
1010 => [1,1,1,1] => [1,1,1,1]
=> 1 = 2 - 1
1011 => [1,1,2] => [2,1,1]
=> 2 = 3 - 1
1100 => [2,2] => [2,2]
=> 2 = 3 - 1
1101 => [2,1,1] => [2,1,1]
=> 2 = 3 - 1
1110 => [3,1] => [3,1]
=> 3 = 4 - 1
1111 => [4] => [4]
=> 4 = 5 - 1
00000 => [5] => [5]
=> 5 = 6 - 1
00001 => [4,1] => [4,1]
=> 4 = 5 - 1
00010 => [3,1,1] => [3,1,1]
=> 3 = 4 - 1
00011 => [3,2] => [3,2]
=> 3 = 4 - 1
00100 => [2,1,2] => [2,2,1]
=> 2 = 3 - 1
00101 => [2,1,1,1] => [2,1,1,1]
=> 2 = 3 - 1
00110 => [2,2,1] => [2,2,1]
=> 2 = 3 - 1
00111 => [2,3] => [3,2]
=> 3 = 4 - 1
01000 => [1,1,3] => [3,1,1]
=> 3 = 4 - 1
01001 => [1,1,2,1] => [2,1,1,1]
=> 2 = 3 - 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> 1 = 2 - 1
01011 => [1,1,1,2] => [2,1,1,1]
=> 2 = 3 - 1
01100 => [1,2,2] => [2,2,1]
=> 2 = 3 - 1
01101 => [1,2,1,1] => [2,1,1,1]
=> 2 = 3 - 1
01110 => [1,3,1] => [3,1,1]
=> 3 = 4 - 1
01111 => [1,4] => [4,1]
=> 4 = 5 - 1
10000 => [1,4] => [4,1]
=> 4 = 5 - 1
10001 => [1,3,1] => [3,1,1]
=> 3 = 4 - 1
10010 => [1,2,1,1] => [2,1,1,1]
=> 2 = 3 - 1
10011 => [1,2,2] => [2,2,1]
=> 2 = 3 - 1
Description
The largest part of an integer partition.
Matching statistic: St001291
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [1,0,1,0]
=> 2
1 => [1] => [1]
=> [1,0,1,0]
=> 2
00 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3
01 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2
10 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2
11 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3
000 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4
001 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3
010 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
011 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3
100 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3
101 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
110 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3
111 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4
0000 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
0001 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
0011 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
0111 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
1000 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
1100 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
1110 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4
1111 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
00000 => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
00001 => [4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
00011 => [3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
00100 => [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
00110 => [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
00111 => [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
01100 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
01111 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
10000 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
10011 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000010
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [1]
=> 1 = 2 - 1
1 => [1] => [1]
=> [1]
=> 1 = 2 - 1
00 => [2] => [2]
=> [1,1]
=> 2 = 3 - 1
01 => [1,1] => [1,1]
=> [2]
=> 1 = 2 - 1
10 => [1,1] => [1,1]
=> [2]
=> 1 = 2 - 1
11 => [2] => [2]
=> [1,1]
=> 2 = 3 - 1
000 => [3] => [3]
=> [1,1,1]
=> 3 = 4 - 1
001 => [2,1] => [2,1]
=> [2,1]
=> 2 = 3 - 1
010 => [1,1,1] => [1,1,1]
=> [3]
=> 1 = 2 - 1
011 => [1,2] => [2,1]
=> [2,1]
=> 2 = 3 - 1
100 => [1,2] => [2,1]
=> [2,1]
=> 2 = 3 - 1
101 => [1,1,1] => [1,1,1]
=> [3]
=> 1 = 2 - 1
110 => [2,1] => [2,1]
=> [2,1]
=> 2 = 3 - 1
111 => [3] => [3]
=> [1,1,1]
=> 3 = 4 - 1
0000 => [4] => [4]
=> [1,1,1,1]
=> 4 = 5 - 1
0001 => [3,1] => [3,1]
=> [2,1,1]
=> 3 = 4 - 1
0010 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2 = 3 - 1
0011 => [2,2] => [2,2]
=> [2,2]
=> 2 = 3 - 1
0100 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2 = 3 - 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1 = 2 - 1
0110 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2 = 3 - 1
0111 => [1,3] => [3,1]
=> [2,1,1]
=> 3 = 4 - 1
1000 => [1,3] => [3,1]
=> [2,1,1]
=> 3 = 4 - 1
1001 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2 = 3 - 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1 = 2 - 1
1011 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2 = 3 - 1
1100 => [2,2] => [2,2]
=> [2,2]
=> 2 = 3 - 1
1101 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2 = 3 - 1
1110 => [3,1] => [3,1]
=> [2,1,1]
=> 3 = 4 - 1
1111 => [4] => [4]
=> [1,1,1,1]
=> 4 = 5 - 1
00000 => [5] => [5]
=> [1,1,1,1,1]
=> 5 = 6 - 1
00001 => [4,1] => [4,1]
=> [2,1,1,1]
=> 4 = 5 - 1
00010 => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
00011 => [3,2] => [3,2]
=> [2,2,1]
=> 3 = 4 - 1
00100 => [2,1,2] => [2,2,1]
=> [3,2]
=> 2 = 3 - 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2 = 3 - 1
00110 => [2,2,1] => [2,2,1]
=> [3,2]
=> 2 = 3 - 1
00111 => [2,3] => [3,2]
=> [2,2,1]
=> 3 = 4 - 1
01000 => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2 = 3 - 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2 = 3 - 1
01100 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2 = 3 - 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2 = 3 - 1
01110 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
01111 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4 = 5 - 1
10000 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4 = 5 - 1
10001 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2 = 3 - 1
10011 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2 = 3 - 1
Description
The length of the partition.
Matching statistic: St000734
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [[1]]
=> 1 = 2 - 1
1 => [1] => [1]
=> [[1]]
=> 1 = 2 - 1
00 => [2] => [2]
=> [[1,2]]
=> 2 = 3 - 1
01 => [1,1] => [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
10 => [1,1] => [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
11 => [2] => [2]
=> [[1,2]]
=> 2 = 3 - 1
000 => [3] => [3]
=> [[1,2,3]]
=> 3 = 4 - 1
001 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
010 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 2 - 1
011 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
100 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
101 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 2 - 1
110 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
111 => [3] => [3]
=> [[1,2,3]]
=> 3 = 4 - 1
0000 => [4] => [4]
=> [[1,2,3,4]]
=> 4 = 5 - 1
0001 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
0010 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
0011 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
0100 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
0110 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
0111 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
1000 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
1001 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
1011 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
1100 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 3 - 1
1101 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
1110 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 4 - 1
1111 => [4] => [4]
=> [[1,2,3,4]]
=> 4 = 5 - 1
00000 => [5] => [5]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
00001 => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 5 - 1
00010 => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
00011 => [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 4 - 1
00100 => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
00110 => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
00111 => [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 4 - 1
01000 => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 2 - 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
01100 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
01110 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
01111 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 5 - 1
10000 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 5 - 1
10001 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
10011 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [1,0]
=> 1 = 2 - 1
1 => [1] => [1]
=> [1,0]
=> 1 = 2 - 1
00 => [2] => [2]
=> [1,0,1,0]
=> 2 = 3 - 1
01 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
10 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
11 => [2] => [2]
=> [1,0,1,0]
=> 2 = 3 - 1
000 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
001 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
010 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
011 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
100 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
101 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
110 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
111 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
0000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
0001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
0011 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
0111 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
1000 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
1100 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
1110 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
1111 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
00000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
00001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
00011 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
00100 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
00110 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
00111 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
01100 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
01111 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
10000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
10011 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
000001010 => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010000010 => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010100000 => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010111110 => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
011111010 => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
100000101 => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101000001 => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101011111 => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101111101 => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
111110101 => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1111101001 => [5,1,1,2,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1101111101 => [2,1,5,1,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1101011111 => [2,1,1,1,5] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1101000001 => [2,1,1,5,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1011111011 => [1,1,5,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1001011111 => [1,2,1,1,5] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1000001101 => [1,5,2,1,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1000001011 => [1,5,1,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1000001001 => [1,5,1,2,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
111000001101 => [3,5,2,1,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
111011000001 => [3,1,2,5,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
100111100001 => [1,2,4,4,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
100001111001 => [1,4,4,2,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
100000110111 => [1,5,2,1,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
100000110001 => [1,5,2,3,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
100000111101 => [1,5,4,1,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101100011111 => [1,1,2,3,5] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
101100000111 => [1,1,2,5,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
101111000001 => [1,1,4,5,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0100111110 => [1,1,2,5,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
000111110010 => [3,5,2,1,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
000100111110 => [3,1,2,5,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
011000011110 => [1,2,4,4,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
011110000110 => [1,4,4,2,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
011111001000 => [1,5,2,1,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
011111000010 => [1,5,4,1,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010011111000 => [1,1,2,5,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
010000111110 => [1,1,4,5,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0000010100 => [5,1,1,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0000010110 => [5,1,1,2,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0010111110 => [2,1,1,5,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
10000100001 => [1,4,1,4,1] => [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
0100000100 => [1,1,5,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
011000001100 => [1,2,5,2,2] => [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
0011111010 => [2,5,1,1,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
01111100100 => [1,5,2,1,2] => [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [1,0]
=> ? = 2 - 1
1 => [1] => [1]
=> [1,0]
=> ? = 2 - 1
00 => [2] => [2]
=> [1,0,1,0]
=> 2 = 3 - 1
01 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
10 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
11 => [2] => [2]
=> [1,0,1,0]
=> 2 = 3 - 1
000 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
001 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
010 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
011 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
100 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
101 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
110 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
111 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
0000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
0001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
0011 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
0111 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
1000 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
1100 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
1110 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
1111 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
00000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
00001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
00011 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
00100 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
00110 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
00111 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
01100 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
01111 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
10000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
10011 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
10100 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
000001010 => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010000010 => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010100000 => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010111110 => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
011111010 => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
100000101 => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101000001 => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101011111 => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101111101 => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
111110101 => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1111101001 => [5,1,1,2,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1101111101 => [2,1,5,1,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1101011111 => [2,1,1,1,5] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1101000001 => [2,1,1,5,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1011111011 => [1,1,5,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1001011111 => [1,2,1,1,5] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1000001101 => [1,5,2,1,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1000001011 => [1,5,1,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
1000001001 => [1,5,1,2,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
111000001101 => [3,5,2,1,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
111011000001 => [3,1,2,5,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
100111100001 => [1,2,4,4,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
100001111001 => [1,4,4,2,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
100000110111 => [1,5,2,1,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
100000110001 => [1,5,2,3,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
100000111101 => [1,5,4,1,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
101100011111 => [1,1,2,3,5] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
101100000111 => [1,1,2,5,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
101111000001 => [1,1,4,5,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0100111110 => [1,1,2,5,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
000111110010 => [3,5,2,1,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
000100111110 => [3,1,2,5,1] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
011000011110 => [1,2,4,4,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
011110000110 => [1,4,4,2,1] => [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 - 1
011111001000 => [1,5,2,1,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
011111000010 => [1,5,4,1,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
010011111000 => [1,1,2,5,3] => [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
010000111110 => [1,1,4,5,1] => [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0000010100 => [5,1,1,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0000010110 => [5,1,1,2,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
0010111110 => [2,1,1,5,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
10000100001 => [1,4,1,4,1] => [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
0100000100 => [1,1,5,1,2] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
011000001100 => [1,2,5,2,2] => [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
0011111010 => [2,5,1,1,1] => [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
01111100100 => [1,5,2,1,2] => [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 6 - 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000982
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00105: Binary words —complement⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
St000982: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 = 2 - 1
1 => 0 => 1 = 2 - 1
00 => 11 => 2 = 3 - 1
01 => 10 => 1 = 2 - 1
10 => 01 => 1 = 2 - 1
11 => 00 => 2 = 3 - 1
000 => 111 => 3 = 4 - 1
001 => 110 => 2 = 3 - 1
010 => 101 => 1 = 2 - 1
011 => 100 => 2 = 3 - 1
100 => 011 => 2 = 3 - 1
101 => 010 => 1 = 2 - 1
110 => 001 => 2 = 3 - 1
111 => 000 => 3 = 4 - 1
0000 => 1111 => 4 = 5 - 1
0001 => 1110 => 3 = 4 - 1
0010 => 1101 => 2 = 3 - 1
0011 => 1100 => 2 = 3 - 1
0100 => 1011 => 2 = 3 - 1
0101 => 1010 => 1 = 2 - 1
0110 => 1001 => 2 = 3 - 1
0111 => 1000 => 3 = 4 - 1
1000 => 0111 => 3 = 4 - 1
1001 => 0110 => 2 = 3 - 1
1010 => 0101 => 1 = 2 - 1
1011 => 0100 => 2 = 3 - 1
1100 => 0011 => 2 = 3 - 1
1101 => 0010 => 2 = 3 - 1
1110 => 0001 => 3 = 4 - 1
1111 => 0000 => 4 = 5 - 1
00000 => 11111 => 5 = 6 - 1
00001 => 11110 => 4 = 5 - 1
00010 => 11101 => 3 = 4 - 1
00011 => 11100 => 3 = 4 - 1
00100 => 11011 => 2 = 3 - 1
00101 => 11010 => 2 = 3 - 1
00110 => 11001 => 2 = 3 - 1
00111 => 11000 => 3 = 4 - 1
01000 => 10111 => 3 = 4 - 1
01001 => 10110 => 2 = 3 - 1
01010 => 10101 => 1 = 2 - 1
01011 => 10100 => 2 = 3 - 1
01100 => 10011 => 2 = 3 - 1
01101 => 10010 => 2 = 3 - 1
01110 => 10001 => 3 = 4 - 1
01111 => 10000 => 4 = 5 - 1
10000 => 01111 => 4 = 5 - 1
10001 => 01110 => 3 = 4 - 1
10010 => 01101 => 2 = 3 - 1
10011 => 01100 => 2 = 3 - 1
1111001111 => ? => ? = 5 - 1
1101000111 => 0010111000 => ? = 4 - 1
1101000001 => 0010111110 => ? = 6 - 1
1100111100 => 0011000011 => ? = 5 - 1
1100111001 => 0011000110 => ? = 4 - 1
1100110001 => 0011001110 => ? = 4 - 1
1100110000 => ? => ? = 5 - 1
1100100111 => 0011011000 => ? = 4 - 1
1100100011 => 0011011100 => ? = 4 - 1
1100011110 => 0011100001 => ? = 5 - 1
1100011101 => 0011100010 => ? = 4 - 1
1100011100 => 0011100011 => ? = 4 - 1
1100011011 => 0011100100 => ? = 4 - 1
1100011000 => 0011100111 => ? = 4 - 1
1100010111 => 0011101000 => ? = 4 - 1
1100001111 => 0011110000 => ? = 5 - 1
1100001100 => 0011110011 => ? = 5 - 1
1100001000 => 0011110111 => ? = 5 - 1
1100000110 => 0011111001 => ? = 6 - 1
1011110111 => 0100001000 => ? = 5 - 1
1011110001 => 0100001110 => ? = 5 - 1
1011100011 => 0100011100 => ? = 4 - 1
1011100001 => 0100011110 => ? = 5 - 1
1011000111 => 0100111000 => ? = 4 - 1
1010001111 => 0101110000 => ? = 5 - 1
1001111000 => 0110000111 => ? = 5 - 1
1001110011 => 0110001100 => ? = 4 - 1
1001110001 => 0110001110 => ? = 4 - 1
1001110000 => 0110001111 => ? = 5 - 1
1001100111 => 0110011000 => ? = 4 - 1
1001011111 => 0110100000 => ? = 6 - 1
1001001111 => 0110110000 => ? = 5 - 1
1000111101 => 0111000010 => ? = 5 - 1
1000111100 => 0111000011 => ? = 5 - 1
1000111011 => 0111000100 => ? = 4 - 1
1000111000 => 0111000111 => ? = 4 - 1
1000110111 => 0111001000 => ? = 4 - 1
1000110001 => 0111001110 => ? = 4 - 1
1000101111 => 0111010000 => ? = 5 - 1
1000011111 => 0111100000 => ? = 6 - 1
1000001101 => 0111110010 => ? = 6 - 1
1000001011 => 0111110100 => ? = 6 - 1
1000001001 => 0111110110 => ? = 6 - 1
110000011110 => 001111100001 => ? = 6 - 1
100001111100 => 011110000011 => ? = 6 - 1
111001100001 => 000110011110 => ? = 5 - 1
111011000001 => 000100111110 => ? = 6 - 1
100111100001 => 011000011110 => ? = 5 - 1
100000110001 => 011111001110 => ? = 6 - 1
100011110001 => 011100001110 => ? = 5 - 1
Description
The length of the longest constant subword.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 => 0 = 2 - 2
1 => [1] => 1 => 0 => 0 = 2 - 2
00 => [2] => 10 => 01 => 1 = 3 - 2
01 => [1,1] => 11 => 00 => 0 = 2 - 2
10 => [1,1] => 11 => 00 => 0 = 2 - 2
11 => [2] => 10 => 01 => 1 = 3 - 2
000 => [3] => 100 => 011 => 2 = 4 - 2
001 => [2,1] => 101 => 010 => 1 = 3 - 2
010 => [1,1,1] => 111 => 000 => 0 = 2 - 2
011 => [1,2] => 110 => 001 => 1 = 3 - 2
100 => [1,2] => 110 => 001 => 1 = 3 - 2
101 => [1,1,1] => 111 => 000 => 0 = 2 - 2
110 => [2,1] => 101 => 010 => 1 = 3 - 2
111 => [3] => 100 => 011 => 2 = 4 - 2
0000 => [4] => 1000 => 0111 => 3 = 5 - 2
0001 => [3,1] => 1001 => 0110 => 2 = 4 - 2
0010 => [2,1,1] => 1011 => 0100 => 1 = 3 - 2
0011 => [2,2] => 1010 => 0101 => 1 = 3 - 2
0100 => [1,1,2] => 1110 => 0001 => 1 = 3 - 2
0101 => [1,1,1,1] => 1111 => 0000 => 0 = 2 - 2
0110 => [1,2,1] => 1101 => 0010 => 1 = 3 - 2
0111 => [1,3] => 1100 => 0011 => 2 = 4 - 2
1000 => [1,3] => 1100 => 0011 => 2 = 4 - 2
1001 => [1,2,1] => 1101 => 0010 => 1 = 3 - 2
1010 => [1,1,1,1] => 1111 => 0000 => 0 = 2 - 2
1011 => [1,1,2] => 1110 => 0001 => 1 = 3 - 2
1100 => [2,2] => 1010 => 0101 => 1 = 3 - 2
1101 => [2,1,1] => 1011 => 0100 => 1 = 3 - 2
1110 => [3,1] => 1001 => 0110 => 2 = 4 - 2
1111 => [4] => 1000 => 0111 => 3 = 5 - 2
00000 => [5] => 10000 => 01111 => 4 = 6 - 2
00001 => [4,1] => 10001 => 01110 => 3 = 5 - 2
00010 => [3,1,1] => 10011 => 01100 => 2 = 4 - 2
00011 => [3,2] => 10010 => 01101 => 2 = 4 - 2
00100 => [2,1,2] => 10110 => 01001 => 1 = 3 - 2
00101 => [2,1,1,1] => 10111 => 01000 => 1 = 3 - 2
00110 => [2,2,1] => 10101 => 01010 => 1 = 3 - 2
00111 => [2,3] => 10100 => 01011 => 2 = 4 - 2
01000 => [1,1,3] => 11100 => 00011 => 2 = 4 - 2
01001 => [1,1,2,1] => 11101 => 00010 => 1 = 3 - 2
01010 => [1,1,1,1,1] => 11111 => 00000 => 0 = 2 - 2
01011 => [1,1,1,2] => 11110 => 00001 => 1 = 3 - 2
01100 => [1,2,2] => 11010 => 00101 => 1 = 3 - 2
01101 => [1,2,1,1] => 11011 => 00100 => 1 = 3 - 2
01110 => [1,3,1] => 11001 => 00110 => 2 = 4 - 2
01111 => [1,4] => 11000 => 00111 => 3 = 5 - 2
10000 => [1,4] => 11000 => 00111 => 3 = 5 - 2
10001 => [1,3,1] => 11001 => 00110 => 2 = 4 - 2
10010 => [1,2,1,1] => 11011 => 00100 => 1 = 3 - 2
10011 => [1,2,2] => 11010 => 00101 => 1 = 3 - 2
1100111000 => [2,2,3,3] => 1010100100 => 0101011011 => ? = 4 - 2
1101110000 => [2,1,3,4] => 1011001000 => 0100110111 => ? = 5 - 2
1110001100 => [3,3,2,2] => 1001001010 => 0110110101 => ? = 4 - 2
1110011000 => [3,2,2,3] => 1001010100 => 0110101011 => ? = 4 - 2
1110110000 => [3,1,2,4] => 1001101000 => 0110010111 => ? = 5 - 2
1111000010 => [4,4,1,1] => 1000100011 => 0111011100 => ? = 5 - 2
1111000100 => [4,3,1,2] => 1000100110 => 0111011001 => ? = 5 - 2
1111001000 => [4,2,1,3] => 1000101100 => 0111010011 => ? = 5 - 2
1111010000 => [4,1,1,4] => 1000111000 => 0111000111 => ? = 5 - 2
110011110000 => [2,2,4,4] => 101010001000 => 010101110111 => ? = 5 - 2
110111100000 => [2,1,4,5] => 101100010000 => 010011101111 => ? = 6 - 2
111001110000 => [3,2,3,4] => 100101001000 => 011010110111 => ? = 5 - 2
111011100000 => [3,1,3,5] => 100110010000 => 011001101111 => ? = 6 - 2
111100001100 => [4,4,2,2] => 100010001010 => 011101110101 => ? = 5 - 2
111100011000 => [4,3,2,3] => 100010010100 => 011101101011 => ? = 5 - 2
111100110000 => [4,2,2,4] => 100010101000 => 011101010111 => ? = 5 - 2
111101100000 => [4,1,2,5] => 100011010000 => 011100101111 => ? = 6 - 2
111110000100 => [5,4,1,2] => 100001000110 => 011110111001 => ? = 6 - 2
111110001000 => [5,3,1,3] => 100001001100 => 011110110011 => ? = 6 - 2
111110010000 => [5,2,1,4] => 100001011000 => 011110100111 => ? = 6 - 2
1111101000 => [5,1,1,3] => 1000011100 => 0111100011 => ? = 6 - 2
1111100011 => [5,3,2] => 1000010010 => 0111101101 => ? = 6 - 2
1111100010 => [5,3,1,1] => 1000010011 => 0111101100 => ? = 6 - 2
1111100001 => [5,4,1] => 1000010001 => 0111101110 => ? = 6 - 2
1111011110 => [4,1,4,1] => 1000110001 => 0111001110 => ? = 5 - 2
1111011000 => [4,1,2,3] => 1000110100 => 0111001011 => ? = 5 - 2
1111010111 => [4,1,1,1,3] => 1000111100 => 0111000011 => ? = 5 - 2
1111001111 => [4,2,4] => 1000101000 => 0111010111 => ? = 5 - 2
1111001100 => [4,2,2,2] => 1000101010 => 0111010101 => ? = 5 - 2
1111000111 => [4,3,3] => 1000100100 => 0111011011 => ? = 5 - 2
1111000110 => [4,3,2,1] => 1000100101 => 0111011010 => ? = 5 - 2
1111000011 => [4,4,2] => 1000100010 => 0111011101 => ? = 5 - 2
1111000001 => [4,5,1] => 1000100001 => 0111011110 => ? = 6 - 2
1110111110 => [3,1,5,1] => 1001100001 => 0110011110 => ? = 6 - 2
1110111000 => [3,1,3,3] => 1001100100 => 0110011011 => ? = 4 - 2
1110101111 => [3,1,1,1,4] => 1001111000 => 0110000111 => ? = 5 - 2
1110100011 => [3,1,1,3,2] => 1001110010 => 0110001101 => ? = 4 - 2
1110100001 => [3,1,1,4,1] => 1001110001 => 0110001110 => ? = 5 - 2
1110011111 => [3,2,5] => 1001010000 => 0110101111 => ? = 6 - 2
1110011110 => [3,2,4,1] => 1001010001 => 0110101110 => ? = 5 - 2
1110000111 => [3,4,3] => 1001000100 => 0110111011 => ? = 5 - 2
1110000011 => [3,5,2] => 1001000010 => 0110111101 => ? = 6 - 2
1110000010 => [3,5,1,1] => 1001000011 => 0110111100 => ? = 6 - 2
1101111011 => [2,1,4,1,2] => 1011000110 => 0100111001 => ? = 5 - 2
1101111000 => [2,1,4,3] => 1011000100 => 0100111011 => ? = 5 - 2
1101110001 => [2,1,3,3,1] => 1011001001 => 0100110110 => ? = 4 - 2
1101100001 => [2,1,2,4,1] => 1011010001 => 0100101110 => ? = 5 - 2
1101000001 => [2,1,1,5,1] => 1011100001 => 0100011110 => ? = 6 - 2
1100111100 => [2,2,4,2] => 1010100010 => 0101011101 => ? = 5 - 2
1100110000 => [2,2,2,4] => 1010101000 => 0101010111 => ? = 5 - 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000381
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 = 2 - 1
1 => [1] => 1 = 2 - 1
00 => [2] => 2 = 3 - 1
01 => [1,1] => 1 = 2 - 1
10 => [1,1] => 1 = 2 - 1
11 => [2] => 2 = 3 - 1
000 => [3] => 3 = 4 - 1
001 => [2,1] => 2 = 3 - 1
010 => [1,1,1] => 1 = 2 - 1
011 => [1,2] => 2 = 3 - 1
100 => [1,2] => 2 = 3 - 1
101 => [1,1,1] => 1 = 2 - 1
110 => [2,1] => 2 = 3 - 1
111 => [3] => 3 = 4 - 1
0000 => [4] => 4 = 5 - 1
0001 => [3,1] => 3 = 4 - 1
0010 => [2,1,1] => 2 = 3 - 1
0011 => [2,2] => 2 = 3 - 1
0100 => [1,1,2] => 2 = 3 - 1
0101 => [1,1,1,1] => 1 = 2 - 1
0110 => [1,2,1] => 2 = 3 - 1
0111 => [1,3] => 3 = 4 - 1
1000 => [1,3] => 3 = 4 - 1
1001 => [1,2,1] => 2 = 3 - 1
1010 => [1,1,1,1] => 1 = 2 - 1
1011 => [1,1,2] => 2 = 3 - 1
1100 => [2,2] => 2 = 3 - 1
1101 => [2,1,1] => 2 = 3 - 1
1110 => [3,1] => 3 = 4 - 1
1111 => [4] => 4 = 5 - 1
00000 => [5] => 5 = 6 - 1
00001 => [4,1] => 4 = 5 - 1
00010 => [3,1,1] => 3 = 4 - 1
00011 => [3,2] => 3 = 4 - 1
00100 => [2,1,2] => 2 = 3 - 1
00101 => [2,1,1,1] => 2 = 3 - 1
00110 => [2,2,1] => 2 = 3 - 1
00111 => [2,3] => 3 = 4 - 1
01000 => [1,1,3] => 3 = 4 - 1
01001 => [1,1,2,1] => 2 = 3 - 1
01010 => [1,1,1,1,1] => 1 = 2 - 1
01011 => [1,1,1,2] => 2 = 3 - 1
01100 => [1,2,2] => 2 = 3 - 1
01101 => [1,2,1,1] => 2 = 3 - 1
01110 => [1,3,1] => 3 = 4 - 1
01111 => [1,4] => 4 = 5 - 1
10000 => [1,4] => 4 = 5 - 1
10001 => [1,3,1] => 3 = 4 - 1
10010 => [1,2,1,1] => 2 = 3 - 1
10011 => [1,2,2] => 2 = 3 - 1
1111101001 => [5,1,1,2,1] => ? = 6 - 1
1111101000 => [5,1,1,3] => ? = 6 - 1
1111100100 => [5,2,1,2] => ? = 6 - 1
1111100011 => [5,3,2] => ? = 6 - 1
1111100010 => [5,3,1,1] => ? = 6 - 1
1111100001 => [5,4,1] => ? = 6 - 1
1111011110 => [4,1,4,1] => ? = 5 - 1
1111011101 => [4,1,3,1,1] => ? = 5 - 1
1111011011 => [4,1,2,1,2] => ? = 5 - 1
1111011000 => [4,1,2,3] => ? = 5 - 1
1111010111 => [4,1,1,1,3] => ? = 5 - 1
1111010011 => [4,1,1,2,2] => ? = 5 - 1
1111010001 => [4,1,1,3,1] => ? = 5 - 1
1111001111 => [4,2,4] => ? = 5 - 1
1111001100 => [4,2,2,2] => ? = 5 - 1
1111001011 => [4,2,1,1,2] => ? = 5 - 1
1111001001 => [4,2,1,2,1] => ? = 5 - 1
1111000111 => [4,3,3] => ? = 5 - 1
1111000110 => [4,3,2,1] => ? = 5 - 1
1111000101 => [4,3,1,1,1] => ? = 5 - 1
1111000011 => [4,4,2] => ? = 5 - 1
1111000001 => [4,5,1] => ? = 6 - 1
1110111110 => [3,1,5,1] => ? = 6 - 1
1110111101 => [3,1,4,1,1] => ? = 5 - 1
1110111011 => [3,1,3,1,2] => ? = 4 - 1
1110111000 => [3,1,3,3] => ? = 4 - 1
1110110001 => [3,1,2,3,1] => ? = 4 - 1
1110101111 => [3,1,1,1,4] => ? = 5 - 1
1110100011 => [3,1,1,3,2] => ? = 4 - 1
1110100001 => [3,1,1,4,1] => ? = 5 - 1
1110011111 => [3,2,5] => ? = 6 - 1
1110011110 => [3,2,4,1] => ? = 5 - 1
1110011001 => [3,2,2,2,1] => ? = 4 - 1
1110010011 => [3,2,1,2,2] => ? = 4 - 1
1110010001 => [3,2,1,3,1] => ? = 4 - 1
1110001101 => [3,3,2,1,1] => ? = 4 - 1
1110001001 => [3,3,1,2,1] => ? = 4 - 1
1110000111 => [3,4,3] => ? = 5 - 1
1110000101 => [3,4,1,1,1] => ? = 5 - 1
1110000011 => [3,5,2] => ? = 6 - 1
1110000010 => [3,5,1,1] => ? = 6 - 1
1101111101 => [2,1,5,1,1] => ? = 6 - 1
1101111011 => [2,1,4,1,2] => ? = 5 - 1
1101111001 => [2,1,4,2,1] => ? = 5 - 1
1101111000 => [2,1,4,3] => ? = 5 - 1
1101110111 => [2,1,3,1,3] => ? = 4 - 1
1101110001 => [2,1,3,3,1] => ? = 4 - 1
1101101111 => [2,1,2,1,4] => ? = 5 - 1
1101100011 => [2,1,2,3,2] => ? = 4 - 1
1101100001 => [2,1,2,4,1] => ? = 5 - 1
Description
The largest part of an integer composition.
Matching statistic: St000983
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 1 = 2 - 1
1 => 1 => 1 => 1 = 2 - 1
00 => 01 => 10 => 2 = 3 - 1
01 => 00 => 00 => 1 = 2 - 1
10 => 11 => 11 => 1 = 2 - 1
11 => 10 => 01 => 2 = 3 - 1
000 => 010 => 010 => 3 = 4 - 1
001 => 011 => 110 => 2 = 3 - 1
010 => 000 => 000 => 1 = 2 - 1
011 => 001 => 100 => 2 = 3 - 1
100 => 110 => 011 => 2 = 3 - 1
101 => 111 => 111 => 1 = 2 - 1
110 => 100 => 001 => 2 = 3 - 1
111 => 101 => 101 => 3 = 4 - 1
0000 => 0101 => 1010 => 4 = 5 - 1
0001 => 0100 => 0010 => 3 = 4 - 1
0010 => 0111 => 1110 => 2 = 3 - 1
0011 => 0110 => 0110 => 2 = 3 - 1
0100 => 0001 => 1000 => 2 = 3 - 1
0101 => 0000 => 0000 => 1 = 2 - 1
0110 => 0011 => 1100 => 2 = 3 - 1
0111 => 0010 => 0100 => 3 = 4 - 1
1000 => 1101 => 1011 => 3 = 4 - 1
1001 => 1100 => 0011 => 2 = 3 - 1
1010 => 1111 => 1111 => 1 = 2 - 1
1011 => 1110 => 0111 => 2 = 3 - 1
1100 => 1001 => 1001 => 2 = 3 - 1
1101 => 1000 => 0001 => 2 = 3 - 1
1110 => 1011 => 1101 => 3 = 4 - 1
1111 => 1010 => 0101 => 4 = 5 - 1
00000 => 01010 => 01010 => 5 = 6 - 1
00001 => 01011 => 11010 => 4 = 5 - 1
00010 => 01000 => 00010 => 3 = 4 - 1
00011 => 01001 => 10010 => 3 = 4 - 1
00100 => 01110 => 01110 => 2 = 3 - 1
00101 => 01111 => 11110 => 2 = 3 - 1
00110 => 01100 => 00110 => 2 = 3 - 1
00111 => 01101 => 10110 => 3 = 4 - 1
01000 => 00010 => 01000 => 3 = 4 - 1
01001 => 00011 => 11000 => 2 = 3 - 1
01010 => 00000 => 00000 => 1 = 2 - 1
01011 => 00001 => 10000 => 2 = 3 - 1
01100 => 00110 => 01100 => 2 = 3 - 1
01101 => 00111 => 11100 => 2 = 3 - 1
01110 => 00100 => 00100 => 3 = 4 - 1
01111 => 00101 => 10100 => 4 = 5 - 1
10000 => 11010 => 01011 => 4 = 5 - 1
10001 => 11011 => 11011 => 3 = 4 - 1
10010 => 11000 => 00011 => 2 = 3 - 1
10011 => 11001 => 10011 => 2 = 3 - 1
1011110000 => 1110100101 => 1010010111 => ? = 5 - 1
1100111000 => 1001101101 => 1011011001 => ? = 4 - 1
1101110000 => 1000100101 => 1010010001 => ? = 5 - 1
1110001100 => 1011011001 => 1001101101 => ? = 4 - 1
1110011000 => 1011001101 => 1011001101 => ? = 4 - 1
1110110000 => 1011100101 => 1010011101 => ? = 5 - 1
1111000010 => 1010010111 => 1110100101 => ? = 5 - 1
1111000100 => 1010010001 => 1000100101 => ? = 5 - 1
1111001000 => 1010011101 => 1011100101 => ? = 5 - 1
1111010000 => 1010000101 => 1010000101 => ? = 5 - 1
110011110000 => 100110100101 => 101001011001 => ? = 5 - 1
110111100000 => 100010110101 => 101011010001 => ? = 6 - 1
111001110000 => 101100100101 => 101001001101 => ? = 5 - 1
111011100000 => 101110110101 => 101011011101 => ? = 6 - 1
111100001100 => 101001011001 => 100110100101 => ? = 5 - 1
111100011000 => 101001001101 => 101100100101 => ? = 5 - 1
111100110000 => 101001100101 => 101001100101 => ? = 5 - 1
111101100000 => 101000110101 => 101011000101 => ? = 6 - 1
111110000100 => 101011010001 => 100010110101 => ? = 6 - 1
111110001000 => 101011011101 => 101110110101 => ? = 6 - 1
111110010000 => 101011000101 => 101000110101 => ? = 6 - 1
1111101001 => 1010111100 => 0011110101 => ? = 6 - 1
1111101000 => 1010111101 => 1011110101 => ? = 6 - 1
1111100100 => 1010110001 => 1000110101 => ? = 6 - 1
1111100011 => 1010110110 => 0110110101 => ? = 6 - 1
1111100010 => 1010110111 => 1110110101 => ? = 6 - 1
1111100001 => 1010110100 => 0010110101 => ? = 6 - 1
1111011110 => 1010001011 => 1101000101 => ? = 5 - 1
1111011011 => 1010001110 => 0111000101 => ? = 5 - 1
1111011000 => 1010001101 => 1011000101 => ? = 5 - 1
1111010111 => 1010000010 => 0100000101 => ? = 5 - 1
1111010011 => 1010000110 => 0110000101 => ? = 5 - 1
1111001111 => 1010011010 => 0101100101 => ? = 5 - 1
1111001100 => 1010011001 => 1001100101 => ? = 5 - 1
1111001011 => 1010011110 => 0111100101 => ? = 5 - 1
1111001001 => 1010011100 => 0011100101 => ? = 5 - 1
1111000111 => 1010010010 => 0100100101 => ? = 5 - 1
1111000110 => 1010010011 => 1100100101 => ? = 5 - 1
1111000011 => 1010010110 => 0110100101 => ? = 5 - 1
1110111110 => 1011101011 => 1101011101 => ? = 6 - 1
1110111011 => 1011101110 => 0111011101 => ? = 4 - 1
1110111000 => 1011101101 => 1011011101 => ? = 4 - 1
1110101111 => 1011111010 => 0101111101 => ? = 5 - 1
1110100011 => 1011110110 => 0110111101 => ? = 4 - 1
1110100001 => 1011110100 => 0010111101 => ? = 5 - 1
1110011111 => 1011001010 => 0101001101 => ? = 6 - 1
1110011110 => 1011001011 => 1101001101 => ? = 5 - 1
1110011001 => 1011001100 => 0011001101 => ? = 4 - 1
1110010011 => 1011000110 => 0110001101 => ? = 4 - 1
1110001001 => 1011011100 => 0011101101 => ? = 4 - 1
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000439The position of the first down step of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000306The bounce count of a Dyck path. St000025The number of initial rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000308The height of the tree associated to a permutation. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000094The depth of an ordered tree. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000028The number of stack-sorts needed to sort a permutation. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St001589The nesting number of a perfect matching. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001330The hat guessing number of a graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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