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Your data matches 76 different statistics following compositions of up to 3 maps.
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Matching statistic: St000982
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(load all 6 compositions to match this statistic)
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 2
01 => 1
10 => 1
11 => 2
000 => 3
001 => 2
010 => 1
011 => 2
100 => 2
101 => 1
110 => 2
111 => 3
0000 => 4
0001 => 3
0010 => 2
0011 => 2
0100 => 2
0101 => 1
0110 => 2
0111 => 3
1000 => 3
1001 => 2
1010 => 1
1011 => 2
1100 => 2
1101 => 2
1110 => 3
1111 => 4
00000 => 5
00001 => 4
00010 => 3
00011 => 3
00100 => 2
00101 => 2
00110 => 2
00111 => 3
01000 => 3
01001 => 2
01010 => 1
01011 => 2
01100 => 2
01101 => 2
01110 => 3
01111 => 4
10000 => 4
10001 => 3
10010 => 2
10011 => 2
Description
The length of the longest constant subword.
Matching statistic: St000147
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> 1
1 => [1] => [1]
=> 1
00 => [2] => [2]
=> 2
01 => [1,1] => [1,1]
=> 1
10 => [1,1] => [1,1]
=> 1
11 => [2] => [2]
=> 2
000 => [3] => [3]
=> 3
001 => [2,1] => [2,1]
=> 2
010 => [1,1,1] => [1,1,1]
=> 1
011 => [1,2] => [2,1]
=> 2
100 => [1,2] => [2,1]
=> 2
101 => [1,1,1] => [1,1,1]
=> 1
110 => [2,1] => [2,1]
=> 2
111 => [3] => [3]
=> 3
0000 => [4] => [4]
=> 4
0001 => [3,1] => [3,1]
=> 3
0010 => [2,1,1] => [2,1,1]
=> 2
0011 => [2,2] => [2,2]
=> 2
0100 => [1,1,2] => [2,1,1]
=> 2
0101 => [1,1,1,1] => [1,1,1,1]
=> 1
0110 => [1,2,1] => [2,1,1]
=> 2
0111 => [1,3] => [3,1]
=> 3
1000 => [1,3] => [3,1]
=> 3
1001 => [1,2,1] => [2,1,1]
=> 2
1010 => [1,1,1,1] => [1,1,1,1]
=> 1
1011 => [1,1,2] => [2,1,1]
=> 2
1100 => [2,2] => [2,2]
=> 2
1101 => [2,1,1] => [2,1,1]
=> 2
1110 => [3,1] => [3,1]
=> 3
1111 => [4] => [4]
=> 4
00000 => [5] => [5]
=> 5
00001 => [4,1] => [4,1]
=> 4
00010 => [3,1,1] => [3,1,1]
=> 3
00011 => [3,2] => [3,2]
=> 3
00100 => [2,1,2] => [2,2,1]
=> 2
00101 => [2,1,1,1] => [2,1,1,1]
=> 2
00110 => [2,2,1] => [2,2,1]
=> 2
00111 => [2,3] => [3,2]
=> 3
01000 => [1,1,3] => [3,1,1]
=> 3
01001 => [1,1,2,1] => [2,1,1,1]
=> 2
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> 1
01011 => [1,1,1,2] => [2,1,1,1]
=> 2
01100 => [1,2,2] => [2,2,1]
=> 2
01101 => [1,2,1,1] => [2,1,1,1]
=> 2
01110 => [1,3,1] => [3,1,1]
=> 3
01111 => [1,4] => [4,1]
=> 4
10000 => [1,4] => [4,1]
=> 4
10001 => [1,3,1] => [3,1,1]
=> 3
10010 => [1,2,1,1] => [2,1,1,1]
=> 2
10011 => [1,2,2] => [2,2,1]
=> 2
10110111100000 => [1,1,2,1,4,5] => ?
=> ? = 5
10111011010000 => [1,1,3,1,2,1,1,4] => ?
=> ? = 4
10111011100000 => [1,1,3,1,3,5] => ?
=> ? = 5
10111100110000 => [1,1,4,2,2,4] => ?
=> ? = 4
10111101001000 => [1,1,4,1,1,2,1,3] => ?
=> ? = 4
10111101010000 => [1,1,4,1,1,1,1,4] => ?
=> ? = 4
10111101100000 => [1,1,4,1,2,5] => ?
=> ? = 5
10111110000100 => [1,1,5,4,1,2] => ?
=> ? = 5
10111110001000 => [1,1,5,3,1,3] => ?
=> ? = 5
10111110010000 => [1,1,5,2,1,4] => ?
=> ? = 5
10111110100000 => [1,1,5,1,1,5] => ?
=> ? = 5
10111111000000 => [1,1,6,6] => ?
=> ? = 6
11001111100000 => [2,2,5,5] => ?
=> ? = 5
11011110000010 => [2,1,4,5,1,1] => ?
=> ? = 5
11101101000010 => [3,1,2,1,1,4,1,1] => ?
=> ? = 4
11101110000010 => [3,1,3,5,1,1] => ?
=> ? = 5
11110011000010 => [4,2,2,4,1,1] => ?
=> ? = 4
11110100100010 => [4,1,1,2,1,3,1,1] => ?
=> ? = 4
11110101000010 => [4,1,1,1,1,4,1,1] => ?
=> ? = 4
11110110000010 => [4,1,2,5,1,1] => ?
=> ? = 5
11111000001100 => [5,5,2,2] => ?
=> ? = 5
11111000010010 => [5,4,1,2,1,1] => ?
=> ? = 5
11111000100010 => [5,3,1,3,1,1] => ?
=> ? = 5
11111001000010 => [5,2,1,4,1,1] => ?
=> ? = 5
11111010000010 => [5,1,1,5,1,1] => ?
=> ? = 5
11111100000010 => [6,6,1,1] => ?
=> ? = 6
1011110111000000 => [1,1,4,1,3,6] => ?
=> ? = 6
1011111010100000 => [1,1,5,1,1,1,1,5] => ?
=> ? = 5
1011111011000000 => [1,1,5,1,2,6] => ?
=> ? = 6
1011111100010000 => [1,1,6,3,1,4] => ?
=> ? = 6
1011111100100000 => [1,1,6,2,1,5] => ?
=> ? = 6
1011111101000000 => [1,1,6,1,1,6] => ?
=> ? = 6
1011111110000000 => [1,1,7,7] => ?
=> ? = 7
1111011100000010 => [4,1,3,6,1,1] => ?
=> ? = 6
1111101010000010 => [5,1,1,1,1,5,1,1] => ?
=> ? = 5
1111101100000010 => [5,1,2,6,1,1] => ?
=> ? = 6
1111110001000010 => [6,3,1,4,1,1] => ?
=> ? = 6
1111110010000010 => [6,2,1,5,1,1] => ?
=> ? = 6
1111110100000010 => [6,1,1,6,1,1] => ?
=> ? = 6
1111111000000010 => [7,7,1,1] => ?
=> ? = 7
10000001010 => [1,6,1,1,1,1] => ?
=> ? = 6
10000010110 => [1,5,1,1,2,1] => ?
=> ? = 5
10000101110 => [1,4,1,1,3,1] => ?
=> ? = 4
10001011110 => [1,3,1,1,4,1] => ?
=> ? = 4
10010111110 => [1,2,1,1,5,1] => ?
=> ? = 5
10101111110 => [1,1,1,1,6,1] => ?
=> ? = 6
10000010000 => [1,5,1,4] => ?
=> ? = 5
10000100010 => [1,4,1,3,1,1] => ?
=> ? = 4
10001000010 => [1,3,1,4,1,1] => ?
=> ? = 4
10000100100 => [1,4,1,2,1,2] => ?
=> ? = 4
Description
The largest part of an integer partition.
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: 78% ●values known / values provided: 78%●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: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [1]
=> 1
1 => [1] => [1]
=> [1]
=> 1
00 => [2] => [2]
=> [1,1]
=> 2
01 => [1,1] => [1,1]
=> [2]
=> 1
10 => [1,1] => [1,1]
=> [2]
=> 1
11 => [2] => [2]
=> [1,1]
=> 2
000 => [3] => [3]
=> [1,1,1]
=> 3
001 => [2,1] => [2,1]
=> [2,1]
=> 2
010 => [1,1,1] => [1,1,1]
=> [3]
=> 1
011 => [1,2] => [2,1]
=> [2,1]
=> 2
100 => [1,2] => [2,1]
=> [2,1]
=> 2
101 => [1,1,1] => [1,1,1]
=> [3]
=> 1
110 => [2,1] => [2,1]
=> [2,1]
=> 2
111 => [3] => [3]
=> [1,1,1]
=> 3
0000 => [4] => [4]
=> [1,1,1,1]
=> 4
0001 => [3,1] => [3,1]
=> [2,1,1]
=> 3
0010 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
0011 => [2,2] => [2,2]
=> [2,2]
=> 2
0100 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
0101 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
0110 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
0111 => [1,3] => [3,1]
=> [2,1,1]
=> 3
1000 => [1,3] => [3,1]
=> [2,1,1]
=> 3
1001 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
1010 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
1011 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
1100 => [2,2] => [2,2]
=> [2,2]
=> 2
1101 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
1110 => [3,1] => [3,1]
=> [2,1,1]
=> 3
1111 => [4] => [4]
=> [1,1,1,1]
=> 4
00000 => [5] => [5]
=> [1,1,1,1,1]
=> 5
00001 => [4,1] => [4,1]
=> [2,1,1,1]
=> 4
00010 => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
00011 => [3,2] => [3,2]
=> [2,2,1]
=> 3
00100 => [2,1,2] => [2,2,1]
=> [3,2]
=> 2
00101 => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
00110 => [2,2,1] => [2,2,1]
=> [3,2]
=> 2
00111 => [2,3] => [3,2]
=> [2,2,1]
=> 3
01000 => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
01001 => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
01100 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2
01101 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
01110 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
01111 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4
10000 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4
10001 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
10010 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
10011 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2
10110111100000 => [1,1,2,1,4,5] => ?
=> ?
=> ? = 5
10111011010000 => [1,1,3,1,2,1,1,4] => ?
=> ?
=> ? = 4
10111011100000 => [1,1,3,1,3,5] => ?
=> ?
=> ? = 5
10111100110000 => [1,1,4,2,2,4] => ?
=> ?
=> ? = 4
10111101001000 => [1,1,4,1,1,2,1,3] => ?
=> ?
=> ? = 4
10111101010000 => [1,1,4,1,1,1,1,4] => ?
=> ?
=> ? = 4
10111101100000 => [1,1,4,1,2,5] => ?
=> ?
=> ? = 5
10111110000100 => [1,1,5,4,1,2] => ?
=> ?
=> ? = 5
10111110001000 => [1,1,5,3,1,3] => ?
=> ?
=> ? = 5
10111110010000 => [1,1,5,2,1,4] => ?
=> ?
=> ? = 5
10111110100000 => [1,1,5,1,1,5] => ?
=> ?
=> ? = 5
10111111000000 => [1,1,6,6] => ?
=> ?
=> ? = 6
11001111100000 => [2,2,5,5] => ?
=> ?
=> ? = 5
11011110000010 => [2,1,4,5,1,1] => ?
=> ?
=> ? = 5
11101101000010 => [3,1,2,1,1,4,1,1] => ?
=> ?
=> ? = 4
11101110000010 => [3,1,3,5,1,1] => ?
=> ?
=> ? = 5
11110011000010 => [4,2,2,4,1,1] => ?
=> ?
=> ? = 4
11110100100010 => [4,1,1,2,1,3,1,1] => ?
=> ?
=> ? = 4
11110101000010 => [4,1,1,1,1,4,1,1] => ?
=> ?
=> ? = 4
11110110000010 => [4,1,2,5,1,1] => ?
=> ?
=> ? = 5
11111000001100 => [5,5,2,2] => ?
=> ?
=> ? = 5
11111000010010 => [5,4,1,2,1,1] => ?
=> ?
=> ? = 5
11111000100010 => [5,3,1,3,1,1] => ?
=> ?
=> ? = 5
11111001000010 => [5,2,1,4,1,1] => ?
=> ?
=> ? = 5
11111010000010 => [5,1,1,5,1,1] => ?
=> ?
=> ? = 5
11111100000010 => [6,6,1,1] => ?
=> ?
=> ? = 6
1011110111000000 => [1,1,4,1,3,6] => ?
=> ?
=> ? = 6
1011111010100000 => [1,1,5,1,1,1,1,5] => ?
=> ?
=> ? = 5
1011111011000000 => [1,1,5,1,2,6] => ?
=> ?
=> ? = 6
1011111100010000 => [1,1,6,3,1,4] => ?
=> ?
=> ? = 6
1011111100100000 => [1,1,6,2,1,5] => ?
=> ?
=> ? = 6
1011111101000000 => [1,1,6,1,1,6] => ?
=> ?
=> ? = 6
1011111110000000 => [1,1,7,7] => ?
=> ?
=> ? = 7
1111011100000010 => [4,1,3,6,1,1] => ?
=> ?
=> ? = 6
1111101010000010 => [5,1,1,1,1,5,1,1] => ?
=> ?
=> ? = 5
1111101100000010 => [5,1,2,6,1,1] => ?
=> ?
=> ? = 6
1111110001000010 => [6,3,1,4,1,1] => ?
=> ?
=> ? = 6
1111110010000010 => [6,2,1,5,1,1] => ?
=> ?
=> ? = 6
1111110100000010 => [6,1,1,6,1,1] => ?
=> ?
=> ? = 6
1111111000000010 => [7,7,1,1] => ?
=> ?
=> ? = 7
10000001010 => [1,6,1,1,1,1] => ?
=> ?
=> ? = 6
10000010110 => [1,5,1,1,2,1] => ?
=> ?
=> ? = 5
10000101110 => [1,4,1,1,3,1] => ?
=> ?
=> ? = 4
10001011110 => [1,3,1,1,4,1] => ?
=> ?
=> ? = 4
10010111110 => [1,2,1,1,5,1] => ?
=> ?
=> ? = 5
10101111110 => [1,1,1,1,6,1] => ?
=> ?
=> ? = 6
10000010000 => [1,5,1,4] => ?
=> ?
=> ? = 5
10000100010 => [1,4,1,3,1,1] => ?
=> ?
=> ? = 4
10001000010 => [1,3,1,4,1,1] => ?
=> ?
=> ? = 4
10000100100 => [1,4,1,2,1,2] => ?
=> ?
=> ? = 4
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: 63% ●values known / values provided: 63%●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: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1]
=> [[1]]
=> 1
1 => [1] => [1]
=> [[1]]
=> 1
00 => [2] => [2]
=> [[1,2]]
=> 2
01 => [1,1] => [1,1]
=> [[1],[2]]
=> 1
10 => [1,1] => [1,1]
=> [[1],[2]]
=> 1
11 => [2] => [2]
=> [[1,2]]
=> 2
000 => [3] => [3]
=> [[1,2,3]]
=> 3
001 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
010 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
011 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
100 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
101 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
110 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
111 => [3] => [3]
=> [[1,2,3]]
=> 3
0000 => [4] => [4]
=> [[1,2,3,4]]
=> 4
0001 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
0010 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
0011 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
0100 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
0101 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
0110 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
0111 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
1000 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
1001 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
1010 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
1011 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
1100 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
1101 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
1110 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
1111 => [4] => [4]
=> [[1,2,3,4]]
=> 4
00000 => [5] => [5]
=> [[1,2,3,4,5]]
=> 5
00001 => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
00010 => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
00011 => [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
00100 => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
00101 => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
00110 => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
00111 => [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
01000 => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
01001 => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
01100 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
01101 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
01110 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
01111 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
10000 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
10001 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
10010 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
10011 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
101010101100 => [1,1,1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101010110010 => [1,1,1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101010110100 => [1,1,1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101010111000 => [1,1,1,1,1,1,3,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
101011001010 => [1,1,1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101011001100 => [1,1,1,1,2,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
101011010010 => [1,1,1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101011010100 => [1,1,1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101011011000 => [1,1,1,1,2,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101011100010 => [1,1,1,1,3,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
101011100100 => [1,1,1,1,3,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101011101000 => [1,1,1,1,3,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
101011110000 => [1,1,1,1,4,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 4
101100101010 => [1,1,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101100101100 => [1,1,2,2,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
101100110010 => [1,1,2,2,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
101100110100 => [1,1,2,2,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
101101001010 => [1,1,2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101101001100 => [1,1,2,1,1,2,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
101101010010 => [1,1,2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101101010100 => [1,1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
101101011000 => [1,1,2,1,1,1,2,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101101100010 => [1,1,2,1,2,3,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101101100100 => [1,1,2,1,2,2,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
101101101000 => [1,1,2,1,2,1,1,3] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101101110000 => [1,1,2,1,3,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
101110001010 => [1,1,3,3,1,1,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
101110010010 => [1,1,3,2,1,2,1,1] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101110010100 => [1,1,3,2,1,1,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101110100010 => [1,1,3,1,1,3,1,1] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
101110100100 => [1,1,3,1,1,2,1,2] => [3,2,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11],[12]]
=> ? = 3
101110101000 => [1,1,3,1,1,1,1,3] => [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 3
101110110000 => [1,1,3,1,2,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
101111000010 => [1,1,4,4,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 4
101111000100 => [1,1,4,3,1,2] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
101111001000 => [1,1,4,2,1,3] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 4
101111010000 => [1,1,4,1,1,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 4
101111100000 => [1,1,5,5] => [5,5,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12]]
=> ? = 5
110010101010 => [2,2,1,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
110010101100 => [2,2,1,1,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
110010110010 => [2,2,1,1,2,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
110010110100 => [2,2,1,1,2,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
110011001010 => [2,2,2,2,1,1,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
110011010010 => [2,2,2,1,1,2,1,1] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
110011010100 => [2,2,2,1,1,1,1,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
110011011000 => [2,2,2,1,2,3] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
110011100100 => [2,2,3,2,1,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 3
110100101010 => [2,1,1,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
110100101100 => [2,1,1,2,1,1,2,2] => [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? = 2
Description
The last entry in the first row of a standard tableau.
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: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 2
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 2
000 => [3] => 3
001 => [2,1] => 2
010 => [1,1,1] => 1
011 => [1,2] => 2
100 => [1,2] => 2
101 => [1,1,1] => 1
110 => [2,1] => 2
111 => [3] => 3
0000 => [4] => 4
0001 => [3,1] => 3
0010 => [2,1,1] => 2
0011 => [2,2] => 2
0100 => [1,1,2] => 2
0101 => [1,1,1,1] => 1
0110 => [1,2,1] => 2
0111 => [1,3] => 3
1000 => [1,3] => 3
1001 => [1,2,1] => 2
1010 => [1,1,1,1] => 1
1011 => [1,1,2] => 2
1100 => [2,2] => 2
1101 => [2,1,1] => 2
1110 => [3,1] => 3
1111 => [4] => 4
00000 => [5] => 5
00001 => [4,1] => 4
00010 => [3,1,1] => 3
00011 => [3,2] => 3
00100 => [2,1,2] => 2
00101 => [2,1,1,1] => 2
00110 => [2,2,1] => 2
00111 => [2,3] => 3
01000 => [1,1,3] => 3
01001 => [1,1,2,1] => 2
01010 => [1,1,1,1,1] => 1
01011 => [1,1,1,2] => 2
01100 => [1,2,2] => 2
01101 => [1,2,1,1] => 2
01110 => [1,3,1] => 3
01111 => [1,4] => 4
10000 => [1,4] => 4
10001 => [1,3,1] => 3
10010 => [1,2,1,1] => 2
10011 => [1,2,2] => 2
10110111100000 => [1,1,2,1,4,5] => ? = 5
10111011010000 => [1,1,3,1,2,1,1,4] => ? = 4
10111011100000 => [1,1,3,1,3,5] => ? = 5
10111100110000 => [1,1,4,2,2,4] => ? = 4
10111101001000 => [1,1,4,1,1,2,1,3] => ? = 4
10111101010000 => [1,1,4,1,1,1,1,4] => ? = 4
10111101100000 => [1,1,4,1,2,5] => ? = 5
10111110000100 => [1,1,5,4,1,2] => ? = 5
10111110001000 => [1,1,5,3,1,3] => ? = 5
10111110010000 => [1,1,5,2,1,4] => ? = 5
10111110100000 => [1,1,5,1,1,5] => ? = 5
10111111000000 => [1,1,6,6] => ? = 6
11001111100000 => [2,2,5,5] => ? = 5
11011110000010 => [2,1,4,5,1,1] => ? = 5
11101101000010 => [3,1,2,1,1,4,1,1] => ? = 4
11101110000010 => [3,1,3,5,1,1] => ? = 5
11110011000010 => [4,2,2,4,1,1] => ? = 4
11110100100010 => [4,1,1,2,1,3,1,1] => ? = 4
11110101000010 => [4,1,1,1,1,4,1,1] => ? = 4
11110110000010 => [4,1,2,5,1,1] => ? = 5
11111000001100 => [5,5,2,2] => ? = 5
11111000010010 => [5,4,1,2,1,1] => ? = 5
11111000100010 => [5,3,1,3,1,1] => ? = 5
11111001000010 => [5,2,1,4,1,1] => ? = 5
11111010000010 => [5,1,1,5,1,1] => ? = 5
11111100000010 => [6,6,1,1] => ? = 6
1011110111000000 => [1,1,4,1,3,6] => ? = 6
1011111010100000 => [1,1,5,1,1,1,1,5] => ? = 5
1011111011000000 => [1,1,5,1,2,6] => ? = 6
1011111100010000 => [1,1,6,3,1,4] => ? = 6
1011111100100000 => [1,1,6,2,1,5] => ? = 6
1011111101000000 => [1,1,6,1,1,6] => ? = 6
1011111110000000 => [1,1,7,7] => ? = 7
1111011100000010 => [4,1,3,6,1,1] => ? = 6
1111101010000010 => [5,1,1,1,1,5,1,1] => ? = 5
1111101100000010 => [5,1,2,6,1,1] => ? = 6
1111110001000010 => [6,3,1,4,1,1] => ? = 6
1111110010000010 => [6,2,1,5,1,1] => ? = 6
1111110100000010 => [6,1,1,6,1,1] => ? = 6
1111111000000010 => [7,7,1,1] => ? = 7
1111000001 => [4,5,1] => ? = 5
1110111110 => [3,1,5,1] => ? = 5
1110100001 => [3,1,1,4,1] => ? = 4
1110011110 => [3,2,4,1] => ? = 4
1110010001 => [3,2,1,3,1] => ? = 3
1110001001 => [3,3,1,2,1] => ? = 3
1110000101 => [3,4,1,1,1] => ? = 4
1101100001 => [2,1,2,4,1] => ? = 4
1101010001 => [2,1,1,1,1,3,1] => ? = 3
1101001110 => [2,1,1,2,3,1] => ? = 3
Description
The largest part of an integer composition.
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: 56% ●values known / values provided: 56%●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: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 => 0 = 1 - 1
1 => [1] => 1 => 0 => 0 = 1 - 1
00 => [2] => 10 => 01 => 1 = 2 - 1
01 => [1,1] => 11 => 00 => 0 = 1 - 1
10 => [1,1] => 11 => 00 => 0 = 1 - 1
11 => [2] => 10 => 01 => 1 = 2 - 1
000 => [3] => 100 => 011 => 2 = 3 - 1
001 => [2,1] => 101 => 010 => 1 = 2 - 1
010 => [1,1,1] => 111 => 000 => 0 = 1 - 1
011 => [1,2] => 110 => 001 => 1 = 2 - 1
100 => [1,2] => 110 => 001 => 1 = 2 - 1
101 => [1,1,1] => 111 => 000 => 0 = 1 - 1
110 => [2,1] => 101 => 010 => 1 = 2 - 1
111 => [3] => 100 => 011 => 2 = 3 - 1
0000 => [4] => 1000 => 0111 => 3 = 4 - 1
0001 => [3,1] => 1001 => 0110 => 2 = 3 - 1
0010 => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
0011 => [2,2] => 1010 => 0101 => 1 = 2 - 1
0100 => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
0101 => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
0110 => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
0111 => [1,3] => 1100 => 0011 => 2 = 3 - 1
1000 => [1,3] => 1100 => 0011 => 2 = 3 - 1
1001 => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
1010 => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
1011 => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
1100 => [2,2] => 1010 => 0101 => 1 = 2 - 1
1101 => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
1110 => [3,1] => 1001 => 0110 => 2 = 3 - 1
1111 => [4] => 1000 => 0111 => 3 = 4 - 1
00000 => [5] => 10000 => 01111 => 4 = 5 - 1
00001 => [4,1] => 10001 => 01110 => 3 = 4 - 1
00010 => [3,1,1] => 10011 => 01100 => 2 = 3 - 1
00011 => [3,2] => 10010 => 01101 => 2 = 3 - 1
00100 => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
00101 => [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
00110 => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
00111 => [2,3] => 10100 => 01011 => 2 = 3 - 1
01000 => [1,1,3] => 11100 => 00011 => 2 = 3 - 1
01001 => [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
01010 => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
01011 => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
01100 => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
01101 => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
01110 => [1,3,1] => 11001 => 00110 => 2 = 3 - 1
01111 => [1,4] => 11000 => 00111 => 3 = 4 - 1
10000 => [1,4] => 11000 => 00111 => 3 = 4 - 1
10001 => [1,3,1] => 11001 => 00110 => 2 = 3 - 1
10010 => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
10011 => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
1100101010 => [2,2,1,1,1,1,1,1] => 1010111111 => 0101000000 => ? = 2 - 1
1100101100 => [2,2,1,1,2,2] => 1010111010 => 0101000101 => ? = 2 - 1
1100110010 => [2,2,2,2,1,1] => 1010101011 => 0101010100 => ? = 2 - 1
1100110100 => [2,2,2,1,1,2] => 1010101110 => 0101010001 => ? = 2 - 1
1100111000 => [2,2,3,3] => 1010100100 => 0101011011 => ? = 3 - 1
1101001010 => [2,1,1,2,1,1,1,1] => 1011101111 => 0100010000 => ? = 2 - 1
1101001100 => [2,1,1,2,2,2] => 1011101010 => 0100010101 => ? = 2 - 1
1101100010 => [2,1,2,3,1,1] => 1011010011 => 0100101100 => ? = 3 - 1
1101100100 => [2,1,2,2,1,2] => 1011010110 => 0100101001 => ? = 2 - 1
1101101000 => [2,1,2,1,1,3] => 1011011100 => 0100100011 => ? = 3 - 1
1101110000 => [2,1,3,4] => 1011001000 => 0100110111 => ? = 4 - 1
1110001010 => [3,3,1,1,1,1] => 1001001111 => 0110110000 => ? = 3 - 1
1110001100 => [3,3,2,2] => 1001001010 => 0110110101 => ? = 3 - 1
1110010010 => [3,2,1,2,1,1] => 1001011011 => 0110100100 => ? = 3 - 1
1110011000 => [3,2,2,3] => 1001010100 => 0110101011 => ? = 3 - 1
1110100010 => [3,1,1,3,1,1] => 1001110011 => 0110001100 => ? = 3 - 1
1110100100 => [3,1,1,2,1,2] => 1001110110 => 0110001001 => ? = 3 - 1
1110101000 => [3,1,1,1,1,3] => 1001111100 => 0110000011 => ? = 3 - 1
1110110000 => [3,1,2,4] => 1001101000 => 0110010111 => ? = 4 - 1
1111000010 => [4,4,1,1] => 1000100011 => 0111011100 => ? = 4 - 1
1111000100 => [4,3,1,2] => 1000100110 => 0111011001 => ? = 4 - 1
1111001000 => [4,2,1,3] => 1000101100 => 0111010011 => ? = 4 - 1
1111010000 => [4,1,1,4] => 1000111000 => 0111000111 => ? = 4 - 1
1111100000 => [5,5] => 1000010000 => 0111101111 => ? = 5 - 1
101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => 111111111111 => 000000000000 => ? = 1 - 1
101010101100 => [1,1,1,1,1,1,1,1,2,2] => 111111111010 => 000000000101 => ? = 2 - 1
101010110010 => [1,1,1,1,1,1,2,2,1,1] => 111111101011 => 000000010100 => ? = 2 - 1
101010110100 => [1,1,1,1,1,1,2,1,1,2] => 111111101110 => 000000010001 => ? = 2 - 1
101010111000 => [1,1,1,1,1,1,3,3] => 111111100100 => 000000011011 => ? = 3 - 1
101011001010 => [1,1,1,1,2,2,1,1,1,1] => 111110101111 => 000001010000 => ? = 2 - 1
101011001100 => [1,1,1,1,2,2,2,2] => 111110101010 => 000001010101 => ? = 2 - 1
101011010010 => [1,1,1,1,2,1,1,2,1,1] => 111110111011 => 000001000100 => ? = 2 - 1
101011010100 => [1,1,1,1,2,1,1,1,1,2] => 111110111110 => 000001000001 => ? = 2 - 1
101011011000 => [1,1,1,1,2,1,2,3] => 111110110100 => 000001001011 => ? = 3 - 1
101011100010 => [1,1,1,1,3,3,1,1] => 111110010011 => 000001101100 => ? = 3 - 1
101011100100 => [1,1,1,1,3,2,1,2] => 111110010110 => 000001101001 => ? = 3 - 1
101011101000 => [1,1,1,1,3,1,1,3] => 111110011100 => 000001100011 => ? = 3 - 1
101100101010 => [1,1,2,2,1,1,1,1,1,1] => 111010111111 => 000101000000 => ? = 2 - 1
101100101100 => [1,1,2,2,1,1,2,2] => 111010111010 => 000101000101 => ? = 2 - 1
101100110010 => [1,1,2,2,2,2,1,1] => 111010101011 => 000101010100 => ? = 2 - 1
101100110100 => [1,1,2,2,2,1,1,2] => 111010101110 => 000101010001 => ? = 2 - 1
101101001010 => [1,1,2,1,1,2,1,1,1,1] => 111011101111 => 000100010000 => ? = 2 - 1
101101001100 => [1,1,2,1,1,2,2,2] => 111011101010 => 000100010101 => ? = 2 - 1
101101010010 => [1,1,2,1,1,1,1,2,1,1] => 111011111011 => 000100000100 => ? = 2 - 1
101101010100 => [1,1,2,1,1,1,1,1,1,2] => 111011111110 => 000100000001 => ? = 2 - 1
101101011000 => [1,1,2,1,1,1,2,3] => 111011110100 => 000100001011 => ? = 3 - 1
101101100010 => [1,1,2,1,2,3,1,1] => 111011010011 => 000100101100 => ? = 3 - 1
101101100100 => [1,1,2,1,2,2,1,2] => 111011010110 => 000100101001 => ? = 2 - 1
101101101000 => [1,1,2,1,2,1,1,3] => 111011011100 => 000100100011 => ? = 3 - 1
101110001010 => [1,1,3,3,1,1,1,1] => 111001001111 => 000110110000 => ? = 3 - 1
Description
The length of the longest run of ones in a binary word.
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: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Values
0 => [1] => [1]
=> [1,0]
=> 1
1 => [1] => [1]
=> [1,0]
=> 1
00 => [2] => [2]
=> [1,0,1,0]
=> 2
01 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
10 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
11 => [2] => [2]
=> [1,0,1,0]
=> 2
000 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
001 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
010 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
011 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
100 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
101 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
110 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
111 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
0000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
0001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
0011 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
0111 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
1000 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
1100 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
1110 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
1111 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
00000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
00001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
00011 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
00100 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
00110 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
00111 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
01100 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
01111 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
10000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
10011 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
01010101 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
10101010 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
000000000 => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
000000001 => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
000000010 => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
000000101 => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
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]
=> ? = 5
000010101 => [4,1,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
000101010 => [3,1,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
001001010 => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010010 => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010100 => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010101 => [2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010110 => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001011010 => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001101010 => [2,2,1,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010000000 => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
010000001 => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
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]
=> ? = 5
010000101 => [1,1,4,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
010001010 => [1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010010010 => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010010100 => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010010101 => [1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010010110 => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010011010 => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
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]
=> ? = 5
010100001 => [1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
010100010 => [1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010100100 => [1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010100101 => [1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010100110 => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101000 => [1,1,1,1,1,1,3] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010101001 => [1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101010 => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
010101011 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101100 => [1,1,1,1,1,2,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101101 => [1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101110 => [1,1,1,1,1,3,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010101111 => [1,1,1,1,1,4] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
010110010 => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010110100 => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010110101 => [1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010110110 => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010111010 => [1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010111101 => [1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
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]
=> ? = 5
010111111 => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
011001010 => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
011010010 => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
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: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 80%
Values
0 => [1] => [1]
=> [1,0]
=> ? = 1
1 => [1] => [1]
=> [1,0]
=> ? = 1
00 => [2] => [2]
=> [1,0,1,0]
=> 2
01 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
10 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
11 => [2] => [2]
=> [1,0,1,0]
=> 2
000 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
001 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
010 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
011 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
100 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
101 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
110 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
111 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
0000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
0001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
0011 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
0111 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
1000 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
1100 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
1110 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
1111 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
00000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
00001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
00011 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
00100 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
00110 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
00111 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
01100 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
01111 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
10000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
10011 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
10100 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
01010101 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
10101010 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
000000000 => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
000000001 => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
000000010 => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
000000101 => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
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]
=> ? = 5
000010101 => [4,1,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
000101010 => [3,1,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
001001010 => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010010 => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010100 => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010101 => [2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001010110 => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001011010 => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
001101010 => [2,2,1,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010000000 => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
010000001 => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
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]
=> ? = 5
010000101 => [1,1,4,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
010001010 => [1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010010010 => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010010100 => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010010101 => [1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010010110 => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010011010 => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
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]
=> ? = 5
010100001 => [1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
010100010 => [1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010100100 => [1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010100101 => [1,1,1,1,2,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010100110 => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101000 => [1,1,1,1,1,1,3] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010101001 => [1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101010 => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
010101011 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101100 => [1,1,1,1,1,2,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101101 => [1,1,1,1,1,2,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010101110 => [1,1,1,1,1,3,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010101111 => [1,1,1,1,1,4] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
010110010 => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010110100 => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010110101 => [1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010110110 => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
010111010 => [1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
010111101 => [1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
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]
=> ? = 5
010111111 => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000983
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 90%
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 90%
Values
0 => 0 => 0 => 1
1 => 1 => 1 => 1
00 => 00 => 01 => 2
01 => 10 => 11 => 1
10 => 01 => 00 => 1
11 => 11 => 10 => 2
000 => 000 => 010 => 3
001 => 100 => 110 => 2
010 => 010 => 000 => 1
011 => 110 => 100 => 2
100 => 001 => 011 => 2
101 => 101 => 111 => 1
110 => 011 => 001 => 2
111 => 111 => 101 => 3
0000 => 0000 => 0101 => 4
0001 => 1000 => 1101 => 3
0010 => 0100 => 0001 => 2
0011 => 1100 => 1001 => 2
0100 => 0010 => 0111 => 2
0101 => 1010 => 1111 => 1
0110 => 0110 => 0011 => 2
0111 => 1110 => 1011 => 3
1000 => 0001 => 0100 => 3
1001 => 1001 => 1100 => 2
1010 => 0101 => 0000 => 1
1011 => 1101 => 1000 => 2
1100 => 0011 => 0110 => 2
1101 => 1011 => 1110 => 2
1110 => 0111 => 0010 => 3
1111 => 1111 => 1010 => 4
00000 => 00000 => 01010 => 5
00001 => 10000 => 11010 => 4
00010 => 01000 => 00010 => 3
00011 => 11000 => 10010 => 3
00100 => 00100 => 01110 => 2
00101 => 10100 => 11110 => 2
00110 => 01100 => 00110 => 2
00111 => 11100 => 10110 => 3
01000 => 00010 => 01000 => 3
01001 => 10010 => 11000 => 2
01010 => 01010 => 00000 => 1
01011 => 11010 => 10000 => 2
01100 => 00110 => 01100 => 2
01101 => 10110 => 11100 => 2
01110 => 01110 => 00100 => 3
01111 => 11110 => 10100 => 4
10000 => 00001 => 01011 => 4
10001 => 10001 => 11011 => 3
10010 => 01001 => 00011 => 2
10011 => 11001 => 10011 => 2
1010101100 => 0011010101 => 0110000000 => ? = 2
1010110100 => 0010110101 => 0111100000 => ? = 2
1010111000 => 0001110101 => 0100100000 => ? = 3
1011001100 => 0011001101 => 0110011000 => ? = 2
1011010100 => 0010101101 => 0111111000 => ? = 2
1011011000 => 0001101101 => 0100111000 => ? = 3
1011100100 => 0010011101 => 0111001000 => ? = 3
1011101000 => 0001011101 => 0100001000 => ? = 3
1011110000 => 0000111101 => 0101101000 => ? = 4
1100101100 => 0011010011 => 0110000110 => ? = 2
1100110100 => 0010110011 => 0111100110 => ? = 2
1100111000 => 0001110011 => 0100100110 => ? = 3
1101001100 => 0011001011 => 0110011110 => ? = 2
1101010100 => 0010101011 => 0111111110 => ? = 2
1101011000 => 0001101011 => 0100111110 => ? = 3
1101100100 => 0010011011 => 0111001110 => ? = 2
1101101000 => 0001011011 => 0100001110 => ? = 3
1101110000 => 0000111011 => 0101101110 => ? = 4
1110001100 => 0011000111 => 0110010010 => ? = 3
1110010100 => 0010100111 => 0111110010 => ? = 3
1110011000 => 0001100111 => 0100110010 => ? = 3
1110100100 => 0010010111 => 0111000010 => ? = 3
1110101000 => 0001010111 => 0100000010 => ? = 3
1110110000 => 0000110111 => 0101100010 => ? = 4
1111000100 => 0010001111 => 0111011010 => ? = 4
1111001000 => 0001001111 => 0100011010 => ? = 4
1111010000 => 0000101111 => 0101111010 => ? = 4
1111100000 => 0000011111 => 0101001010 => ? = 5
101010101010 => 010101010101 => 000000000000 => ? = 1
101010101100 => 001101010101 => 011000000000 => ? = 2
101010110010 => 010011010101 => 000110000000 => ? = 2
101010110100 => 001011010101 => 011110000000 => ? = 2
101010111000 => 000111010101 => 010010000000 => ? = 3
101011001010 => 010100110101 => 000001100000 => ? = 2
101011001100 => 001100110101 => 011001100000 => ? = 2
101011010010 => 010010110101 => 000111100000 => ? = 2
101011010100 => 001010110101 => 011111100000 => ? = 2
101011011000 => 000110110101 => 010011100000 => ? = 3
101011100010 => 010001110101 => 000100100000 => ? = 3
101011100100 => 001001110101 => 011100100000 => ? = 3
101011101000 => 000101110101 => 010000100000 => ? = 3
101011110000 => 000011110101 => 010110100000 => ? = 4
101100101010 => 010101001101 => 000000011000 => ? = 2
101100101100 => 001101001101 => 011000011000 => ? = 2
101100110010 => 010011001101 => 000110011000 => ? = 2
101100110100 => 001011001101 => 011110011000 => ? = 2
101100111000 => 000111001101 => 010010011000 => ? = 3
101101001010 => 010100101101 => 000001111000 => ? = 2
101101001100 => 001100101101 => 011001111000 => ? = 2
101101010010 => 010010101101 => 000111111000 => ? = 2
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 $101...$.
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: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1]
=> [1,0,1,0]
=> 2 = 1 + 1
1 => [1] => [1]
=> [1,0,1,0]
=> 2 = 1 + 1
00 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
01 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
10 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
11 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
000 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
001 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
010 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
011 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
100 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
101 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
110 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
111 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
0000 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
0001 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
0011 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
0111 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
1000 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
1100 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
1110 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
1111 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
00000 => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
00001 => [4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
00011 => [3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
00100 => [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
00110 => [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
00111 => [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
01100 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
01111 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
10000 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
10011 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
000000 => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
111111 => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
0000000 => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
0000001 => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
0010101 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0100101 => [1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0101001 => [1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0101010 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
0101011 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0101101 => [1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0110101 => [1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0111111 => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
1000000 => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
1001010 => [1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
1010010 => [1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
1010100 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
1010101 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
1010110 => [1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
1011010 => [1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
1101010 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
1111110 => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
1111111 => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
00000000 => [8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 + 1
00000001 => [7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 + 1
00000010 => [6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
00000011 => [6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
00010101 => [3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
00100101 => [2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
00101001 => [2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
00101010 => [2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
00101011 => [2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
00101101 => [2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
00110101 => [2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
00111111 => [2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 1
01000000 => [1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
01000101 => [1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
01001001 => [1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
01001010 => [1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
01001011 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
01001101 => [1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
01010001 => [1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
01010010 => [1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
01010011 => [1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
01010100 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
01010101 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
01010110 => [1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
01010111 => [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
01011001 => [1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
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 $A$.
The following 66 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. St000306The bounce count of 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. St000662The staircase size of the code of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000141The maximum drop size of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. 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:
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. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. 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. St000094The depth of an ordered tree. 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. 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. St001330The hat guessing number of a graph. St001717The largest size of an interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000080The rank of the poset. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000272The treewidth of a graph. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000778The metric dimension of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001270The bandwidth of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001962The proper pathwidth of a graph. St000379The number of Hamiltonian cycles in a graph. St000636The hull number of a graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001580The acyclic chromatic number of a graph. St001638The book thickness of a graph.
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