searching the database
Your data matches 76 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
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 = 0 + 1
1 => [1] => [1]
=> 1 = 0 + 1
00 => [2] => [2]
=> 2 = 1 + 1
01 => [1,1] => [1,1]
=> 1 = 0 + 1
10 => [1,1] => [1,1]
=> 1 = 0 + 1
11 => [2] => [2]
=> 2 = 1 + 1
000 => [3] => [3]
=> 3 = 2 + 1
001 => [2,1] => [2,1]
=> 2 = 1 + 1
010 => [1,1,1] => [1,1,1]
=> 1 = 0 + 1
011 => [1,2] => [2,1]
=> 2 = 1 + 1
100 => [1,2] => [2,1]
=> 2 = 1 + 1
101 => [1,1,1] => [1,1,1]
=> 1 = 0 + 1
110 => [2,1] => [2,1]
=> 2 = 1 + 1
111 => [3] => [3]
=> 3 = 2 + 1
0000 => [4] => [4]
=> 4 = 3 + 1
0001 => [3,1] => [3,1]
=> 3 = 2 + 1
0010 => [2,1,1] => [2,1,1]
=> 2 = 1 + 1
0011 => [2,2] => [2,2]
=> 2 = 1 + 1
0100 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,1,1,1]
=> 1 = 0 + 1
0110 => [1,2,1] => [2,1,1]
=> 2 = 1 + 1
0111 => [1,3] => [3,1]
=> 3 = 2 + 1
1000 => [1,3] => [3,1]
=> 3 = 2 + 1
1001 => [1,2,1] => [2,1,1]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
=> 1 = 0 + 1
1011 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
1100 => [2,2] => [2,2]
=> 2 = 1 + 1
1101 => [2,1,1] => [2,1,1]
=> 2 = 1 + 1
1110 => [3,1] => [3,1]
=> 3 = 2 + 1
1111 => [4] => [4]
=> 4 = 3 + 1
00000 => [5] => [5]
=> 5 = 4 + 1
00001 => [4,1] => [4,1]
=> 4 = 3 + 1
00010 => [3,1,1] => [3,1,1]
=> 3 = 2 + 1
00011 => [3,2] => [3,2]
=> 3 = 2 + 1
00100 => [2,1,2] => [2,2,1]
=> 2 = 1 + 1
00101 => [2,1,1,1] => [2,1,1,1]
=> 2 = 1 + 1
00110 => [2,2,1] => [2,2,1]
=> 2 = 1 + 1
00111 => [2,3] => [3,2]
=> 3 = 2 + 1
01000 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
01001 => [1,1,2,1] => [2,1,1,1]
=> 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> 1 = 0 + 1
01011 => [1,1,1,2] => [2,1,1,1]
=> 2 = 1 + 1
01100 => [1,2,2] => [2,2,1]
=> 2 = 1 + 1
01101 => [1,2,1,1] => [2,1,1,1]
=> 2 = 1 + 1
01110 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
01111 => [1,4] => [4,1]
=> 4 = 3 + 1
10000 => [1,4] => [4,1]
=> 4 = 3 + 1
10001 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
10010 => [1,2,1,1] => [2,1,1,1]
=> 2 = 1 + 1
10011 => [1,2,2] => [2,2,1]
=> 2 = 1 + 1
Description
The largest part of an integer partition.
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 0 => 0
1 => [1] => 1 => 0 => 0
00 => [2] => 10 => 01 => 1
01 => [1,1] => 11 => 00 => 0
10 => [1,1] => 11 => 00 => 0
11 => [2] => 10 => 01 => 1
000 => [3] => 100 => 011 => 2
001 => [2,1] => 101 => 010 => 1
010 => [1,1,1] => 111 => 000 => 0
011 => [1,2] => 110 => 001 => 1
100 => [1,2] => 110 => 001 => 1
101 => [1,1,1] => 111 => 000 => 0
110 => [2,1] => 101 => 010 => 1
111 => [3] => 100 => 011 => 2
0000 => [4] => 1000 => 0111 => 3
0001 => [3,1] => 1001 => 0110 => 2
0010 => [2,1,1] => 1011 => 0100 => 1
0011 => [2,2] => 1010 => 0101 => 1
0100 => [1,1,2] => 1110 => 0001 => 1
0101 => [1,1,1,1] => 1111 => 0000 => 0
0110 => [1,2,1] => 1101 => 0010 => 1
0111 => [1,3] => 1100 => 0011 => 2
1000 => [1,3] => 1100 => 0011 => 2
1001 => [1,2,1] => 1101 => 0010 => 1
1010 => [1,1,1,1] => 1111 => 0000 => 0
1011 => [1,1,2] => 1110 => 0001 => 1
1100 => [2,2] => 1010 => 0101 => 1
1101 => [2,1,1] => 1011 => 0100 => 1
1110 => [3,1] => 1001 => 0110 => 2
1111 => [4] => 1000 => 0111 => 3
00000 => [5] => 10000 => 01111 => 4
00001 => [4,1] => 10001 => 01110 => 3
00010 => [3,1,1] => 10011 => 01100 => 2
00011 => [3,2] => 10010 => 01101 => 2
00100 => [2,1,2] => 10110 => 01001 => 1
00101 => [2,1,1,1] => 10111 => 01000 => 1
00110 => [2,2,1] => 10101 => 01010 => 1
00111 => [2,3] => 10100 => 01011 => 2
01000 => [1,1,3] => 11100 => 00011 => 2
01001 => [1,1,2,1] => 11101 => 00010 => 1
01010 => [1,1,1,1,1] => 11111 => 00000 => 0
01011 => [1,1,1,2] => 11110 => 00001 => 1
01100 => [1,2,2] => 11010 => 00101 => 1
01101 => [1,2,1,1] => 11011 => 00100 => 1
01110 => [1,3,1] => 11001 => 00110 => 2
01111 => [1,4] => 11000 => 00111 => 3
10000 => [1,4] => 11000 => 00111 => 3
10001 => [1,3,1] => 11001 => 00110 => 2
10010 => [1,2,1,1] => 11011 => 00100 => 1
10011 => [1,2,2] => 11010 => 00101 => 1
Description
The length of the longest run of ones in a binary word.
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 = 0 + 1
1 => [1] => [1]
=> [1]
=> 1 = 0 + 1
00 => [2] => [2]
=> [1,1]
=> 2 = 1 + 1
01 => [1,1] => [1,1]
=> [2]
=> 1 = 0 + 1
10 => [1,1] => [1,1]
=> [2]
=> 1 = 0 + 1
11 => [2] => [2]
=> [1,1]
=> 2 = 1 + 1
000 => [3] => [3]
=> [1,1,1]
=> 3 = 2 + 1
001 => [2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
010 => [1,1,1] => [1,1,1]
=> [3]
=> 1 = 0 + 1
011 => [1,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
100 => [1,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
101 => [1,1,1] => [1,1,1]
=> [3]
=> 1 = 0 + 1
110 => [2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
111 => [3] => [3]
=> [1,1,1]
=> 3 = 2 + 1
0000 => [4] => [4]
=> [1,1,1,1]
=> 4 = 3 + 1
0001 => [3,1] => [3,1]
=> [2,1,1]
=> 3 = 2 + 1
0010 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
0011 => [2,2] => [2,2]
=> [2,2]
=> 2 = 1 + 1
0100 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
0110 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
0111 => [1,3] => [3,1]
=> [2,1,1]
=> 3 = 2 + 1
1000 => [1,3] => [3,1]
=> [2,1,1]
=> 3 = 2 + 1
1001 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
1011 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
1100 => [2,2] => [2,2]
=> [2,2]
=> 2 = 1 + 1
1101 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2 = 1 + 1
1110 => [3,1] => [3,1]
=> [2,1,1]
=> 3 = 2 + 1
1111 => [4] => [4]
=> [1,1,1,1]
=> 4 = 3 + 1
00000 => [5] => [5]
=> [1,1,1,1,1]
=> 5 = 4 + 1
00001 => [4,1] => [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
00010 => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
00011 => [3,2] => [3,2]
=> [2,2,1]
=> 3 = 2 + 1
00100 => [2,1,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
00110 => [2,2,1] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
00111 => [2,3] => [3,2]
=> [2,2,1]
=> 3 = 2 + 1
01000 => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
01100 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
01110 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
01111 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
10000 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
10001 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
10011 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2 = 1 + 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: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Values
0 => [1] => [1]
=> [[1]]
=> 1 = 0 + 1
1 => [1] => [1]
=> [[1]]
=> 1 = 0 + 1
00 => [2] => [2]
=> [[1,2]]
=> 2 = 1 + 1
01 => [1,1] => [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
10 => [1,1] => [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
11 => [2] => [2]
=> [[1,2]]
=> 2 = 1 + 1
000 => [3] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
001 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
010 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
011 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
100 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
101 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
110 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
111 => [3] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
0000 => [4] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
0001 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
0010 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
0011 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
0100 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
0110 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
0111 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
1000 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
1001 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
1011 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
1100 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
1101 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
1110 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
1111 => [4] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
00000 => [5] => [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
00001 => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
00010 => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
00011 => [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 2 + 1
00100 => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
00110 => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
00111 => [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 2 + 1
01000 => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
01100 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
01110 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
01111 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
10000 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
10001 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
10011 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
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]]
=> ? = 3 + 1
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]]
=> ? = 3 + 1
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]]
=> ? = 3 + 1
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]]
=> ? = 3 + 1
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]]
=> ? = 3 + 1
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]]
=> ? = 3 + 1
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]]
=> ? = 2 + 1
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]]
=> ? = 2 + 1
110101110000 => [2,1,1,1,3,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
110110001100 => [2,1,2,3,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
110110011000 => [2,1,2,2,2,3] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
110110110000 => [2,1,2,1,2,4] => [4,2,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 3 + 1
110111000010 => [2,1,3,4,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
110111010000 => [2,1,3,1,1,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
111001001100 => [3,2,1,2,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
111001100100 => [3,2,2,2,1,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
111011000010 => [3,1,2,4,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
111100001010 => [4,4,1,1,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 3 + 1
111100010010 => [4,3,1,2,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
111100010100 => [4,3,1,1,1,2] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
111100100010 => [4,2,1,3,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
111100100100 => [4,2,1,2,1,2] => [4,2,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 3 + 1
111101000010 => [4,1,1,4,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 3 + 1
111101000100 => [4,1,1,3,1,2] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
010100001111 => [1,1,1,1,4,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 3 + 1
010010001111 => [1,1,2,1,3,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
010000110111 => [1,1,4,2,1,3] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
010000101111 => [1,1,4,1,1,4] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 3 + 1
001100100111 => [2,2,2,1,2,3] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
001100011011 => [2,2,3,2,1,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
001010001111 => [2,1,1,1,3,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
001000101111 => [2,1,3,1,1,4] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
000110110011 => [3,2,1,2,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
000110011011 => [3,2,2,2,1,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
000011110101 => [4,4,1,1,1,1] => [4,4,1,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11],[12]]
=> ? = 3 + 1
000011011101 => [4,2,1,3,1,1] => [4,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11],[12]]
=> ? = 3 + 1
000011011011 => [4,2,1,2,1,2] => [4,2,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 3 + 1
00000000000 => [11] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
10100100101 => [1,1,1,2,1,2,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
10100101001 => [1,1,1,2,1,1,1,2,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
10010100101 => [1,2,1,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
01001010010 => [1,1,2,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
01010101001 => [1,1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
100111001100 => [1,2,3,2,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
011100110011 => [1,3,2,2,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
11111111111 => [11] => [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
01101011010 => [1,2,1,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1 + 1
00000000011 => [9,2] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 8 + 1
11000000000 => [2,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 8 + 1
011000110011 => [1,2,3,2,2,2] => [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 2 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000982
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Mp00105: Binary words —complement⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 91%
Values
0 => 0 => 1 => 1 = 0 + 1
1 => 1 => 0 => 1 = 0 + 1
00 => 00 => 11 => 2 = 1 + 1
01 => 10 => 01 => 1 = 0 + 1
10 => 01 => 10 => 1 = 0 + 1
11 => 11 => 00 => 2 = 1 + 1
000 => 000 => 111 => 3 = 2 + 1
001 => 100 => 011 => 2 = 1 + 1
010 => 010 => 101 => 1 = 0 + 1
011 => 110 => 001 => 2 = 1 + 1
100 => 001 => 110 => 2 = 1 + 1
101 => 101 => 010 => 1 = 0 + 1
110 => 011 => 100 => 2 = 1 + 1
111 => 111 => 000 => 3 = 2 + 1
0000 => 0000 => 1111 => 4 = 3 + 1
0001 => 1000 => 0111 => 3 = 2 + 1
0010 => 0100 => 1011 => 2 = 1 + 1
0011 => 1100 => 0011 => 2 = 1 + 1
0100 => 0010 => 1101 => 2 = 1 + 1
0101 => 1010 => 0101 => 1 = 0 + 1
0110 => 0110 => 1001 => 2 = 1 + 1
0111 => 1110 => 0001 => 3 = 2 + 1
1000 => 0001 => 1110 => 3 = 2 + 1
1001 => 1001 => 0110 => 2 = 1 + 1
1010 => 0101 => 1010 => 1 = 0 + 1
1011 => 1101 => 0010 => 2 = 1 + 1
1100 => 0011 => 1100 => 2 = 1 + 1
1101 => 1011 => 0100 => 2 = 1 + 1
1110 => 0111 => 1000 => 3 = 2 + 1
1111 => 1111 => 0000 => 4 = 3 + 1
00000 => 00000 => 11111 => 5 = 4 + 1
00001 => 10000 => 01111 => 4 = 3 + 1
00010 => 01000 => 10111 => 3 = 2 + 1
00011 => 11000 => 00111 => 3 = 2 + 1
00100 => 00100 => 11011 => 2 = 1 + 1
00101 => 10100 => 01011 => 2 = 1 + 1
00110 => 01100 => 10011 => 2 = 1 + 1
00111 => 11100 => 00011 => 3 = 2 + 1
01000 => 00010 => 11101 => 3 = 2 + 1
01001 => 10010 => 01101 => 2 = 1 + 1
01010 => 01010 => 10101 => 1 = 0 + 1
01011 => 11010 => 00101 => 2 = 1 + 1
01100 => 00110 => 11001 => 2 = 1 + 1
01101 => 10110 => 01001 => 2 = 1 + 1
01110 => 01110 => 10001 => 3 = 2 + 1
01111 => 11110 => 00001 => 4 = 3 + 1
10000 => 00001 => 11110 => 4 = 3 + 1
10001 => 10001 => 01110 => 3 = 2 + 1
10010 => 01001 => 10110 => 2 = 1 + 1
10011 => 11001 => 00110 => 2 = 1 + 1
1111111100 => 0011111111 => 1100000000 => ? = 7 + 1
1111110101 => 1010111111 => 0101000000 => ? = 5 + 1
1111110011 => 1100111111 => 0011000000 => ? = 5 + 1
1111101001 => 1001011111 => 0110100000 => ? = 4 + 1
1111100100 => 0010011111 => 1101100000 => ? = 4 + 1
1111011101 => 1011101111 => 0100010000 => ? = 3 + 1
1111010100 => 0010101111 => 1101010000 => ? = 3 + 1
1111010011 => 1100101111 => 0011010000 => ? = 3 + 1
1111010001 => 1000101111 => 0111010000 => ? = 3 + 1
1111001001 => 1001001111 => 0110110000 => ? = 3 + 1
1111000101 => 1010001111 => 0101110000 => ? = 3 + 1
1110111101 => 1011110111 => 0100001000 => ? = 3 + 1
1110110001 => 1000110111 => 0111001000 => ? = 2 + 1
1110011001 => 1001100111 => 0110011000 => ? = 2 + 1
1110010011 => 1100100111 => 0011011000 => ? = 2 + 1
1110001101 => 1011000111 => 0100111000 => ? = 2 + 1
1101111001 => 1001111011 => 0110000100 => ? = 3 + 1
1101100011 => 1100011011 => 0011100100 => ? = 2 + 1
1101100000 => 0000011011 => 1111100100 => ? = 4 + 1
1101010101 => 1010101011 => 0101010100 => ? = 1 + 1
1101010011 => 1100101011 => 0011010100 => ? = 1 + 1
1100111001 => 1001110011 => 0110001100 => ? = 2 + 1
1100110011 => 1100110011 => 0011001100 => ? = 1 + 1
1100011101 => 1011100011 => 0100011100 => ? = 2 + 1
1100010011 => 1100100011 => 0011011100 => ? = 2 + 1
1100000011 => 1100000011 => 0011111100 => ? = 5 + 1
1100000000 => 0000000011 => 1111111100 => ? = 7 + 1
1011111100 => 0011111101 => 1100000010 => ? = 5 + 1
1011111010 => 0101111101 => 1010000010 => ? = 4 + 1
1011111000 => 0001111101 => 1110000010 => ? = 4 + 1
1011110100 => 0010111101 => 1101000010 => ? = 3 + 1
1011110001 => 1000111101 => 0111000010 => ? = 3 + 1
1011100011 => 1100011101 => 0011100010 => ? = 2 + 1
1011100000 => 0000011101 => 1111100010 => ? = 4 + 1
1011011001 => 1001101101 => 0110010010 => ? = 1 + 1
1011010101 => 1010101101 => 0101010010 => ? = 1 + 1
1011010011 => 1100101101 => 0011010010 => ? = 1 + 1
1011010000 => 0000101101 => 1111010010 => ? = 3 + 1
1011001101 => 1011001101 => 0100110010 => ? = 1 + 1
1011001011 => 1101001101 => 0010110010 => ? = 1 + 1
1011001000 => 0001001101 => 1110110010 => ? = 2 + 1
1011000100 => 0010001101 => 1101110010 => ? = 2 + 1
1011000010 => 0100001101 => 1011110010 => ? = 3 + 1
1011000000 => 0000001101 => 1111110010 => ? = 5 + 1
1010110101 => 1010110101 => 0101001010 => ? = 1 + 1
1010110011 => 1100110101 => 0011001010 => ? = 1 + 1
1010110000 => 0000110101 => 1111001010 => ? = 3 + 1
1010101101 => 1011010101 => 0100101010 => ? = 1 + 1
1010101000 => 0001010101 => 1110101010 => ? = 2 + 1
1010100100 => 0010010101 => 1101101010 => ? = 1 + 1
Description
The length of the longest constant subword.
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
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 82%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 82%
Values
0 => [1] => [1] => 1 = 0 + 1
1 => [1] => [1] => 1 = 0 + 1
00 => [2] => [2] => 2 = 1 + 1
01 => [1,1] => [1,1] => 1 = 0 + 1
10 => [1,1] => [1,1] => 1 = 0 + 1
11 => [2] => [2] => 2 = 1 + 1
000 => [3] => [3] => 3 = 2 + 1
001 => [2,1] => [1,2] => 2 = 1 + 1
010 => [1,1,1] => [1,1,1] => 1 = 0 + 1
011 => [1,2] => [2,1] => 2 = 1 + 1
100 => [1,2] => [2,1] => 2 = 1 + 1
101 => [1,1,1] => [1,1,1] => 1 = 0 + 1
110 => [2,1] => [1,2] => 2 = 1 + 1
111 => [3] => [3] => 3 = 2 + 1
0000 => [4] => [4] => 4 = 3 + 1
0001 => [3,1] => [1,3] => 3 = 2 + 1
0010 => [2,1,1] => [1,1,2] => 2 = 1 + 1
0011 => [2,2] => [2,2] => 2 = 1 + 1
0100 => [1,1,2] => [2,1,1] => 2 = 1 + 1
0101 => [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
0110 => [1,2,1] => [1,2,1] => 2 = 1 + 1
0111 => [1,3] => [3,1] => 3 = 2 + 1
1000 => [1,3] => [3,1] => 3 = 2 + 1
1001 => [1,2,1] => [1,2,1] => 2 = 1 + 1
1010 => [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
1011 => [1,1,2] => [2,1,1] => 2 = 1 + 1
1100 => [2,2] => [2,2] => 2 = 1 + 1
1101 => [2,1,1] => [1,1,2] => 2 = 1 + 1
1110 => [3,1] => [1,3] => 3 = 2 + 1
1111 => [4] => [4] => 4 = 3 + 1
00000 => [5] => [5] => 5 = 4 + 1
00001 => [4,1] => [1,4] => 4 = 3 + 1
00010 => [3,1,1] => [1,1,3] => 3 = 2 + 1
00011 => [3,2] => [2,3] => 3 = 2 + 1
00100 => [2,1,2] => [2,1,2] => 2 = 1 + 1
00101 => [2,1,1,1] => [1,1,1,2] => 2 = 1 + 1
00110 => [2,2,1] => [1,2,2] => 2 = 1 + 1
00111 => [2,3] => [3,2] => 3 = 2 + 1
01000 => [1,1,3] => [3,1,1] => 3 = 2 + 1
01001 => [1,1,2,1] => [1,2,1,1] => 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1] => 1 = 0 + 1
01011 => [1,1,1,2] => [2,1,1,1] => 2 = 1 + 1
01100 => [1,2,2] => [2,2,1] => 2 = 1 + 1
01101 => [1,2,1,1] => [1,1,2,1] => 2 = 1 + 1
01110 => [1,3,1] => [1,3,1] => 3 = 2 + 1
01111 => [1,4] => [4,1] => 4 = 3 + 1
10000 => [1,4] => [4,1] => 4 = 3 + 1
10001 => [1,3,1] => [1,3,1] => 3 = 2 + 1
10010 => [1,2,1,1] => [1,1,2,1] => 2 = 1 + 1
10011 => [1,2,2] => [2,2,1] => 2 = 1 + 1
1111111111 => [10] => [10] => ? = 9 + 1
1111111100 => [8,2] => [2,8] => ? = 7 + 1
1111111011 => [7,1,2] => [2,1,7] => ? = 6 + 1
1111110101 => [6,1,1,1,1] => [1,1,1,1,6] => ? = 5 + 1
1111110011 => [6,2,2] => [2,2,6] => ? = 5 + 1
1111101001 => [5,1,1,2,1] => [1,2,1,1,5] => ? = 4 + 1
1111100100 => [5,2,1,2] => [2,1,2,5] => ? = 4 + 1
1111011101 => [4,1,3,1,1] => [1,1,3,1,4] => ? = 3 + 1
1111011011 => [4,1,2,1,2] => [2,1,2,1,4] => ? = 3 + 1
1111010100 => [4,1,1,1,1,2] => [2,1,1,1,1,4] => ? = 3 + 1
1111010011 => [4,1,1,2,2] => [2,2,1,1,4] => ? = 3 + 1
1111010001 => [4,1,1,3,1] => [1,3,1,1,4] => ? = 3 + 1
1111001011 => [4,2,1,1,2] => [2,1,1,2,4] => ? = 3 + 1
1111001001 => [4,2,1,2,1] => [1,2,1,2,4] => ? = 3 + 1
1111000101 => [4,3,1,1,1] => [1,1,1,3,4] => ? = 3 + 1
1110111101 => [3,1,4,1,1] => [1,1,4,1,3] => ? = 3 + 1
1110111011 => [3,1,3,1,2] => [2,1,3,1,3] => ? = 2 + 1
1110110001 => [3,1,2,3,1] => [1,3,2,1,3] => ? = 2 + 1
1110101011 => [3,1,1,1,1,1,2] => [2,1,1,1,1,1,3] => ? = 2 + 1
1110011001 => [3,2,2,2,1] => [1,2,2,2,3] => ? = 2 + 1
1110010011 => [3,2,1,2,2] => [2,2,1,2,3] => ? = 2 + 1
1110010001 => [3,2,1,3,1] => [1,3,1,2,3] => ? = 2 + 1
1110001101 => [3,3,2,1,1] => [1,1,2,3,3] => ? = 2 + 1
1110001001 => [3,3,1,2,1] => [1,2,1,3,3] => ? = 2 + 1
1110000101 => [3,4,1,1,1] => [1,1,1,4,3] => ? = 3 + 1
1101111111 => [2,1,7] => [7,1,2] => ? = 6 + 1
1101111101 => [2,1,5,1,1] => [1,1,5,1,2] => ? = 4 + 1
1101111001 => [2,1,4,2,1] => [1,2,4,1,2] => ? = 3 + 1
1101110111 => [2,1,3,1,3] => [3,1,3,1,2] => ? = 2 + 1
1101101111 => [2,1,2,1,4] => [4,1,2,1,2] => ? = 3 + 1
1101100011 => [2,1,2,3,2] => [2,3,2,1,2] => ? = 2 + 1
1101100000 => [2,1,2,5] => [5,2,1,2] => ? = 4 + 1
1101011111 => [2,1,1,1,5] => [5,1,1,1,2] => ? = 4 + 1
1101010111 => [2,1,1,1,1,1,3] => [3,1,1,1,1,1,2] => ? = 2 + 1
1101010101 => [2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2] => ? = 1 + 1
1101010011 => [2,1,1,1,1,2,2] => [2,2,1,1,1,1,2] => ? = 1 + 1
1101001111 => [2,1,1,2,4] => [4,2,1,1,2] => ? = 3 + 1
1101000111 => [2,1,1,3,3] => [3,3,1,1,2] => ? = 2 + 1
1101000011 => [2,1,1,4,2] => [2,4,1,1,2] => ? = 3 + 1
1100111111 => [2,2,6] => [6,2,2] => ? = 5 + 1
1100111001 => [2,2,3,2,1] => [1,2,3,2,2] => ? = 2 + 1
1100110011 => [2,2,2,2,2] => [2,2,2,2,2] => ? = 1 + 1
1100110001 => [2,2,2,3,1] => [1,3,2,2,2] => ? = 2 + 1
1100101011 => [2,2,1,1,1,1,2] => [2,1,1,1,1,2,2] => ? = 1 + 1
1100100111 => [2,2,1,2,3] => [3,2,1,2,2] => ? = 2 + 1
1100100011 => [2,2,1,3,2] => [2,3,1,2,2] => ? = 2 + 1
1100011101 => [2,3,3,1,1] => [1,1,3,3,2] => ? = 2 + 1
1100011011 => [2,3,2,1,2] => [2,1,2,3,2] => ? = 2 + 1
1100011001 => [2,3,2,2,1] => [1,2,2,3,2] => ? = 2 + 1
1100010011 => [2,3,1,2,2] => [2,2,1,3,2] => ? = 2 + 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
St000983: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 82%
St000983: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 82%
Values
0 => 0 => 1 = 0 + 1
1 => 1 => 1 = 0 + 1
00 => 01 => 2 = 1 + 1
01 => 00 => 1 = 0 + 1
10 => 11 => 1 = 0 + 1
11 => 10 => 2 = 1 + 1
000 => 010 => 3 = 2 + 1
001 => 011 => 2 = 1 + 1
010 => 000 => 1 = 0 + 1
011 => 001 => 2 = 1 + 1
100 => 110 => 2 = 1 + 1
101 => 111 => 1 = 0 + 1
110 => 100 => 2 = 1 + 1
111 => 101 => 3 = 2 + 1
0000 => 0101 => 4 = 3 + 1
0001 => 0100 => 3 = 2 + 1
0010 => 0111 => 2 = 1 + 1
0011 => 0110 => 2 = 1 + 1
0100 => 0001 => 2 = 1 + 1
0101 => 0000 => 1 = 0 + 1
0110 => 0011 => 2 = 1 + 1
0111 => 0010 => 3 = 2 + 1
1000 => 1101 => 3 = 2 + 1
1001 => 1100 => 2 = 1 + 1
1010 => 1111 => 1 = 0 + 1
1011 => 1110 => 2 = 1 + 1
1100 => 1001 => 2 = 1 + 1
1101 => 1000 => 2 = 1 + 1
1110 => 1011 => 3 = 2 + 1
1111 => 1010 => 4 = 3 + 1
00000 => 01010 => 5 = 4 + 1
00001 => 01011 => 4 = 3 + 1
00010 => 01000 => 3 = 2 + 1
00011 => 01001 => 3 = 2 + 1
00100 => 01110 => 2 = 1 + 1
00101 => 01111 => 2 = 1 + 1
00110 => 01100 => 2 = 1 + 1
00111 => 01101 => 3 = 2 + 1
01000 => 00010 => 3 = 2 + 1
01001 => 00011 => 2 = 1 + 1
01010 => 00000 => 1 = 0 + 1
01011 => 00001 => 2 = 1 + 1
01100 => 00110 => 2 = 1 + 1
01101 => 00111 => 2 = 1 + 1
01110 => 00100 => 3 = 2 + 1
01111 => 00101 => 4 = 3 + 1
10000 => 11010 => 4 = 3 + 1
10001 => 11011 => 3 = 2 + 1
10010 => 11000 => 2 = 1 + 1
10011 => 11001 => 2 = 1 + 1
1010101010 => 1111111111 => ? = 0 + 1
1010101100 => 1111111001 => ? = 1 + 1
1010110010 => 1111100111 => ? = 1 + 1
1010110100 => 1111100001 => ? = 1 + 1
1010111000 => 1111101101 => ? = 2 + 1
1011001010 => 1110011111 => ? = 1 + 1
1011001100 => 1110011001 => ? = 1 + 1
1011010010 => 1110000111 => ? = 1 + 1
1011010100 => 1110000001 => ? = 1 + 1
1011011000 => 1110001101 => ? = 2 + 1
1011100010 => 1110110111 => ? = 2 + 1
1011100100 => 1110110001 => ? = 2 + 1
1011101000 => 1110111101 => ? = 2 + 1
1011110000 => 1110100101 => ? = 3 + 1
1101010010 => 1000000111 => ? = 1 + 1
1101010100 => 1000000001 => ? = 1 + 1
1101011000 => 1000001101 => ? = 2 + 1
1110010100 => 1011000001 => ? = 2 + 1
101011110000 => 111110100101 => ? = 3 + 1
101100111000 => 111001101101 => ? = 2 + 1
101101110000 => 111000100101 => ? = 3 + 1
101110001100 => 111011011001 => ? = 2 + 1
101110011000 => 111011001101 => ? = 2 + 1
101110110000 => 111011100101 => ? = 3 + 1
101111000010 => 111010010111 => ? = 3 + 1
101111001000 => 111010011101 => ? = 3 + 1
101111010000 => 111010000101 => ? = 3 + 1
110010111000 => 100111101101 => ? = 2 + 1
110011001100 => 100110011001 => ? = 1 + 1
110011011000 => 100110001101 => ? = 2 + 1
110011100010 => 100110110111 => ? = 2 + 1
110011100100 => 100110110001 => ? = 2 + 1
110100111000 => 100001101101 => ? = 2 + 1
110101110000 => 100000100101 => ? = 3 + 1
110110001100 => 100011011001 => ? = 2 + 1
110110011000 => 100011001101 => ? = 2 + 1
110110110000 => 100011100101 => ? = 3 + 1
110111000010 => 100010010111 => ? = 3 + 1
110111001000 => 100010011101 => ? = 2 + 1
110111010000 => 100010000101 => ? = 3 + 1
111000101100 => 101101111001 => ? = 2 + 1
111000110010 => 101101100111 => ? = 2 + 1
111000110100 => 101101100001 => ? = 2 + 1
111001001100 => 101100011001 => ? = 2 + 1
111001100010 => 101100110111 => ? = 2 + 1
111001100100 => 101100110001 => ? = 2 + 1
111011000010 => 101110010111 => ? = 3 + 1
111011000100 => 101110010001 => ? = 2 + 1
111100001010 => 101001011111 => ? = 3 + 1
111100010010 => 101001000111 => ? = 3 + 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 $101...$.
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: 73% ●values known / values provided: 74%●distinct values known / distinct values provided: 73%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 74%●distinct values known / distinct values provided: 73%
Values
0 => [1] => [1]
=> [1,0]
=> 1 = 0 + 1
1 => [1] => [1]
=> [1,0]
=> 1 = 0 + 1
00 => [2] => [2]
=> [1,0,1,0]
=> 2 = 1 + 1
01 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
10 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
11 => [2] => [2]
=> [1,0,1,0]
=> 2 = 1 + 1
000 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
001 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
011 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
100 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
110 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
111 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
0000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
0001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
0011 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
0111 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1000 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
1100 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
1110 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1111 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
00000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
00001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
00011 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
00100 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
00110 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
00111 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
01100 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
01111 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
10000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
10011 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 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]
=> ? = 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]
=> ? = 0 + 1
000000000 => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
000000001 => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 + 1
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]
=> ? = 6 + 1
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]
=> ? = 5 + 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]
=> ? = 4 + 1
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]
=> ? = 3 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 6 + 1
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]
=> ? = 5 + 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]
=> ? = 4 + 1
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]
=> ? = 3 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 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]
=> ? = 4 + 1
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]
=> ? = 3 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 2 + 1
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]
=> ? = 3 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 2 + 1
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]
=> ? = 3 + 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]
=> ? = 4 + 1
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]
=> ? = 5 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 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: 73% ●values known / values provided: 74%●distinct values known / distinct values provided: 73%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 74%●distinct values known / distinct values provided: 73%
Values
0 => [1] => [1]
=> [1,0]
=> ? = 0 + 1
1 => [1] => [1]
=> [1,0]
=> ? = 0 + 1
00 => [2] => [2]
=> [1,0,1,0]
=> 2 = 1 + 1
01 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
10 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
11 => [2] => [2]
=> [1,0,1,0]
=> 2 = 1 + 1
000 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
001 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
011 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
100 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
110 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
111 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
0000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
0001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
0011 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
0111 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1000 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
1100 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
1110 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1111 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
00000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
00001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
00011 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
00100 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
00110 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
00111 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
01100 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
01111 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
10000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
10011 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
10100 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 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]
=> ? = 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]
=> ? = 0 + 1
000000000 => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
000000001 => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 + 1
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]
=> ? = 6 + 1
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]
=> ? = 5 + 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]
=> ? = 4 + 1
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]
=> ? = 3 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 6 + 1
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]
=> ? = 5 + 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]
=> ? = 4 + 1
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]
=> ? = 3 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 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]
=> ? = 4 + 1
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]
=> ? = 3 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 2 + 1
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]
=> ? = 1 + 1
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]
=> ? = 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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 2 + 1
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]
=> ? = 3 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 1 + 1
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]
=> ? = 2 + 1
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]
=> ? = 3 + 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]
=> ? = 4 + 1
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]
=> ? = 5 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
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: 45% ●values known / values provided: 56%●distinct values known / distinct values provided: 45%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 56%●distinct values known / distinct values provided: 45%
Values
0 => [1] => [1]
=> [1,0,1,0]
=> 2 = 0 + 2
1 => [1] => [1]
=> [1,0,1,0]
=> 2 = 0 + 2
00 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
01 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
10 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
11 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
000 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
001 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
010 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
011 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
100 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
101 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
110 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
111 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
0000 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 3 + 2
0001 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
0010 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
0011 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
0100 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
0110 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
0111 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
1000 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
1001 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
1011 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
1100 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
1101 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 1 + 2
1110 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 2 + 2
1111 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 3 + 2
00000 => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 4 + 2
00001 => [4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 3 + 2
00010 => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
00011 => [3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
00100 => [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
00101 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
00110 => [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
00111 => [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 2 + 2
01000 => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
01001 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
01010 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 0 + 2
01011 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
01100 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
01101 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
01110 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
01111 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 3 + 2
10000 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 3 + 2
10001 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 2 + 2
10010 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
10011 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
000000 => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 + 2
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]
=> ? = 0 + 2
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]
=> ? = 0 + 2
111111 => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5 + 2
0000000 => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 + 2
0000001 => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 0 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
0111111 => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
1000000 => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 0 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
1111110 => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
1111111 => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 + 2
00000000 => [8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7 + 2
00000001 => [7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 2
00000010 => [6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
00000011 => [6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 5 + 2
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]
=> ? = 2 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
00111111 => [2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 5 + 2
01000000 => [1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 5 + 2
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]
=> ? = 2 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 2 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 1 + 2
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]
=> ? = 0 + 2
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]
=> ? = 1 + 2
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]
=> ? = 2 + 2
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]
=> ? = 1 + 2
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. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise 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. St000094The depth of an ordered tree. St000503The maximal difference between two elements in a common block. St001062The maximal size of a block of a set partition. 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. St000956The maximal displacement of a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000308The height of the tree associated to a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. 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. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. 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. 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 $K[x]/(x^n)$. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St000379The number of Hamiltonian cycles in 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. St001638The book thickness of a graph. 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. St000636The hull number of a graph. St001580The acyclic chromatic number of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!