searching the database
Your data matches 12 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: St000982
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 2
[2] => 10 => 1
[1,1,1] => 111 => 3
[1,2] => 110 => 2
[2,1] => 101 => 1
[3] => 100 => 2
[1,1,1,1] => 1111 => 4
[1,1,2] => 1110 => 3
[1,2,1] => 1101 => 2
[1,3] => 1100 => 2
[2,1,1] => 1011 => 2
[2,2] => 1010 => 1
[3,1] => 1001 => 2
[4] => 1000 => 3
[1,1,1,1,1] => 11111 => 5
[1,1,1,2] => 11110 => 4
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 3
[1,2,1,1] => 11011 => 2
[1,2,2] => 11010 => 2
[1,3,1] => 11001 => 2
[1,4] => 11000 => 3
[2,1,1,1] => 10111 => 3
[2,1,2] => 10110 => 2
[2,2,1] => 10101 => 1
[2,3] => 10100 => 2
[3,1,1] => 10011 => 2
[3,2] => 10010 => 2
[4,1] => 10001 => 3
[5] => 10000 => 4
[1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => 111110 => 5
[1,1,1,2,1] => 111101 => 4
[1,1,1,3] => 111100 => 4
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 3
[1,1,3,1] => 111001 => 3
[1,1,4] => 111000 => 3
[1,2,1,1,1] => 110111 => 3
[1,2,1,2] => 110110 => 2
[1,2,2,1] => 110101 => 2
[1,2,3] => 110100 => 2
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 2
[1,4,1] => 110001 => 3
[1,5] => 110000 => 4
[2,1,1,1,1] => 101111 => 4
[2,1,1,2] => 101110 => 3
[2,1,2,1] => 101101 => 2
Description
The length of the longest constant subword.
Matching statistic: St000147
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1] => [1]
=> 1
[1,1] => 11 => [2] => [2]
=> 2
[2] => 10 => [1,1] => [1,1]
=> 1
[1,1,1] => 111 => [3] => [3]
=> 3
[1,2] => 110 => [2,1] => [2,1]
=> 2
[2,1] => 101 => [1,1,1] => [1,1,1]
=> 1
[3] => 100 => [1,2] => [2,1]
=> 2
[1,1,1,1] => 1111 => [4] => [4]
=> 4
[1,1,2] => 1110 => [3,1] => [3,1]
=> 3
[1,2,1] => 1101 => [2,1,1] => [2,1,1]
=> 2
[1,3] => 1100 => [2,2] => [2,2]
=> 2
[2,1,1] => 1011 => [1,1,2] => [2,1,1]
=> 2
[2,2] => 1010 => [1,1,1,1] => [1,1,1,1]
=> 1
[3,1] => 1001 => [1,2,1] => [2,1,1]
=> 2
[4] => 1000 => [1,3] => [3,1]
=> 3
[1,1,1,1,1] => 11111 => [5] => [5]
=> 5
[1,1,1,2] => 11110 => [4,1] => [4,1]
=> 4
[1,1,2,1] => 11101 => [3,1,1] => [3,1,1]
=> 3
[1,1,3] => 11100 => [3,2] => [3,2]
=> 3
[1,2,1,1] => 11011 => [2,1,2] => [2,2,1]
=> 2
[1,2,2] => 11010 => [2,1,1,1] => [2,1,1,1]
=> 2
[1,3,1] => 11001 => [2,2,1] => [2,2,1]
=> 2
[1,4] => 11000 => [2,3] => [3,2]
=> 3
[2,1,1,1] => 10111 => [1,1,3] => [3,1,1]
=> 3
[2,1,2] => 10110 => [1,1,2,1] => [2,1,1,1]
=> 2
[2,2,1] => 10101 => [1,1,1,1,1] => [1,1,1,1,1]
=> 1
[2,3] => 10100 => [1,1,1,2] => [2,1,1,1]
=> 2
[3,1,1] => 10011 => [1,2,2] => [2,2,1]
=> 2
[3,2] => 10010 => [1,2,1,1] => [2,1,1,1]
=> 2
[4,1] => 10001 => [1,3,1] => [3,1,1]
=> 3
[5] => 10000 => [1,4] => [4,1]
=> 4
[1,1,1,1,1,1] => 111111 => [6] => [6]
=> 6
[1,1,1,1,2] => 111110 => [5,1] => [5,1]
=> 5
[1,1,1,2,1] => 111101 => [4,1,1] => [4,1,1]
=> 4
[1,1,1,3] => 111100 => [4,2] => [4,2]
=> 4
[1,1,2,1,1] => 111011 => [3,1,2] => [3,2,1]
=> 3
[1,1,2,2] => 111010 => [3,1,1,1] => [3,1,1,1]
=> 3
[1,1,3,1] => 111001 => [3,2,1] => [3,2,1]
=> 3
[1,1,4] => 111000 => [3,3] => [3,3]
=> 3
[1,2,1,1,1] => 110111 => [2,1,3] => [3,2,1]
=> 3
[1,2,1,2] => 110110 => [2,1,2,1] => [2,2,1,1]
=> 2
[1,2,2,1] => 110101 => [2,1,1,1,1] => [2,1,1,1,1]
=> 2
[1,2,3] => 110100 => [2,1,1,2] => [2,2,1,1]
=> 2
[1,3,1,1] => 110011 => [2,2,2] => [2,2,2]
=> 2
[1,3,2] => 110010 => [2,2,1,1] => [2,2,1,1]
=> 2
[1,4,1] => 110001 => [2,3,1] => [3,2,1]
=> 3
[1,5] => 110000 => [2,4] => [4,2]
=> 4
[2,1,1,1,1] => 101111 => [1,1,4] => [4,1,1]
=> 4
[2,1,1,2] => 101110 => [1,1,3,1] => [3,1,1,1]
=> 3
[2,1,2,1] => 101101 => [1,1,2,1,1] => [2,1,1,1,1]
=> 2
[3,2,1,1,2,3] => 100101110100 => [1,2,1,1,3,1,1,2] => ?
=> ? = 3
[3,1,1,2,2,3] => 100111010100 => [1,2,3,1,1,1,1,2] => ?
=> ? = 3
[3,1,1,1,2,4] => 100111101000 => [1,2,4,1,1,3] => ?
=> ? = 4
[4,1,1,2,1,3] => 100011101100 => [1,3,3,1,2,2] => ?
=> ? = 3
[1,2,2,3,3,1] => 110101001001 => [2,1,1,1,1,2,1,2,1] => ?
=> ? = 2
[1,3,3,2,2,1] => 110010010101 => [2,2,1,2,1,1,1,1,1] => ?
=> ? = 2
[1,3,2,2,3,1] => 110010101001 => [2,2,1,1,1,1,1,2,1] => ?
=> ? = 2
[3,3,3,2] => 10010010010 => [1,2,1,2,1,2,1,1] => ?
=> ? = 2
[2,3,3,3] => 10100100100 => [1,1,1,2,1,2,1,2] => ?
=> ? = 2
[4,4,3] => 10001000100 => [1,3,1,3,1,2] => ?
=> ? = 3
[2,2,1,1,1,1,1,2] => 10101111110 => [1,1,1,1,6,1] => ?
=> ? = 6
[1,2,2,4,2,1] => 110101000101 => [2,1,1,1,1,3,1,1,1] => ?
=> ? = 3
[1,2,4,2,2,1] => 110100010101 => [2,1,1,3,1,1,1,1,1] => ?
=> ? = 3
[1,4,2,2,2,1] => 110001010101 => [2,3,1,1,1,1,1,1,1] => ?
=> ? = 3
[1,1,1,1,1,1,1,1,2,1] => 11111111101 => [9,1,1] => ?
=> ? = 9
Description
The largest part of an integer partition.
Matching statistic: St000381
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1] => 1
[1,1] => 11 => [2] => 2
[2] => 10 => [1,1] => 1
[1,1,1] => 111 => [3] => 3
[1,2] => 110 => [2,1] => 2
[2,1] => 101 => [1,1,1] => 1
[3] => 100 => [1,2] => 2
[1,1,1,1] => 1111 => [4] => 4
[1,1,2] => 1110 => [3,1] => 3
[1,2,1] => 1101 => [2,1,1] => 2
[1,3] => 1100 => [2,2] => 2
[2,1,1] => 1011 => [1,1,2] => 2
[2,2] => 1010 => [1,1,1,1] => 1
[3,1] => 1001 => [1,2,1] => 2
[4] => 1000 => [1,3] => 3
[1,1,1,1,1] => 11111 => [5] => 5
[1,1,1,2] => 11110 => [4,1] => 4
[1,1,2,1] => 11101 => [3,1,1] => 3
[1,1,3] => 11100 => [3,2] => 3
[1,2,1,1] => 11011 => [2,1,2] => 2
[1,2,2] => 11010 => [2,1,1,1] => 2
[1,3,1] => 11001 => [2,2,1] => 2
[1,4] => 11000 => [2,3] => 3
[2,1,1,1] => 10111 => [1,1,3] => 3
[2,1,2] => 10110 => [1,1,2,1] => 2
[2,2,1] => 10101 => [1,1,1,1,1] => 1
[2,3] => 10100 => [1,1,1,2] => 2
[3,1,1] => 10011 => [1,2,2] => 2
[3,2] => 10010 => [1,2,1,1] => 2
[4,1] => 10001 => [1,3,1] => 3
[5] => 10000 => [1,4] => 4
[1,1,1,1,1,1] => 111111 => [6] => 6
[1,1,1,1,2] => 111110 => [5,1] => 5
[1,1,1,2,1] => 111101 => [4,1,1] => 4
[1,1,1,3] => 111100 => [4,2] => 4
[1,1,2,1,1] => 111011 => [3,1,2] => 3
[1,1,2,2] => 111010 => [3,1,1,1] => 3
[1,1,3,1] => 111001 => [3,2,1] => 3
[1,1,4] => 111000 => [3,3] => 3
[1,2,1,1,1] => 110111 => [2,1,3] => 3
[1,2,1,2] => 110110 => [2,1,2,1] => 2
[1,2,2,1] => 110101 => [2,1,1,1,1] => 2
[1,2,3] => 110100 => [2,1,1,2] => 2
[1,3,1,1] => 110011 => [2,2,2] => 2
[1,3,2] => 110010 => [2,2,1,1] => 2
[1,4,1] => 110001 => [2,3,1] => 3
[1,5] => 110000 => [2,4] => 4
[2,1,1,1,1] => 101111 => [1,1,4] => 4
[2,1,1,2] => 101110 => [1,1,3,1] => 3
[2,1,2,1] => 101101 => [1,1,2,1,1] => 2
[1,1,1,6,1] => 1111000001 => [4,5,1] => ? = 5
[1,1,2,1,1,1,1,2] => 1110111110 => [3,1,5,1] => ? = 5
[1,1,2,5,1] => 1110100001 => [3,1,1,4,1] => ? = 4
[1,1,3,1,1,1,2] => 1110011110 => [3,2,4,1] => ? = 4
[1,1,3,4,1] => 1110010001 => [3,2,1,3,1] => ? = 3
[1,1,4,3,1] => 1110001001 => [3,3,1,2,1] => ? = 3
[1,1,5,2,1] => 1110000101 => [3,4,1,1,1] => ? = 4
[1,2,1,5,1] => 1101100001 => [2,1,2,4,1] => ? = 4
[1,2,2,4,1] => 1101010001 => [2,1,1,1,1,3,1] => ? = 3
[1,2,3,1,1,2] => 1101001110 => [2,1,1,2,3,1] => ? = 3
[1,2,3,3,1] => 1101001001 => [2,1,1,2,1,2,1] => ? = 2
[1,2,4,2,1] => 1101000101 => [2,1,1,3,1,1,1] => ? = 3
[1,3,1,4,1] => 1100110001 => [2,2,2,3,1] => ? = 3
[1,3,2,3,1] => 1100101001 => [2,2,1,1,1,2,1] => ? = 2
[1,3,3,1,2] => 1100100110 => [2,2,1,2,2,1] => ? = 2
[1,3,3,2,1] => 1100100101 => [2,2,1,2,1,1,1] => ? = 2
[1,4,1,3,1] => 1100011001 => [2,3,2,2,1] => ? = 3
[1,4,2,2,1] => 1100010101 => [2,3,1,1,1,1,1] => ? = 3
[1,4,3,2] => 1100010010 => [2,3,1,2,1,1] => ? = 3
[1,5,4] => 1100001000 => [2,4,1,3] => ? = 4
[2,1,1,1,1,1,2,1] => 1011111101 => [1,1,6,1,1] => ? = 6
[2,1,1,5,1] => 1011100001 => [1,1,3,4,1] => ? = 4
[2,1,2,4,1] => 1011010001 => [1,1,2,1,1,3,1] => ? = 3
[2,1,3,3,1] => 1011001001 => [1,1,2,2,1,2,1] => ? = 2
[2,1,4,2,1] => 1011000101 => [1,1,2,3,1,1,1] => ? = 3
[2,2,1,4,1] => 1010110001 => [1,1,1,1,2,3,1] => ? = 3
[2,2,2,3,1] => 1010101001 => [1,1,1,1,1,1,1,2,1] => ? = 2
[2,2,3,1,2] => 1010100110 => [1,1,1,1,1,2,2,1] => ? = 2
[2,2,3,2,1] => 1010100101 => [1,1,1,1,1,2,1,1,1] => ? = 2
[2,3,1,2,2] => 1010011010 => [1,1,1,2,2,1,1,1] => ? = 2
[2,3,1,3,1] => 1010011001 => [1,1,1,2,2,2,1] => ? = 2
[2,3,2,1,2] => 1010010110 => [1,1,1,2,1,1,2,1] => ? = 2
[2,3,2,2,1] => 1010010101 => [1,1,1,2,1,1,1,1,1] => ? = 2
[2,4,1,1,2] => 1010001110 => [1,1,1,3,3,1] => ? = 3
[3,1,1,1,4] => 1001111000 => [1,2,4,3] => ? = 4
[3,1,1,2,3] => 1001110100 => [1,2,3,1,1,2] => ? = 3
[3,1,1,4,1] => 1001110001 => [1,2,3,3,1] => ? = 3
[3,1,2,1,3] => 1001101100 => [1,2,2,1,2,2] => ? = 2
[3,1,2,2,2] => 1001101010 => [1,2,2,1,1,1,1,1] => ? = 2
[3,1,2,3,1] => 1001101001 => [1,2,2,1,1,2,1] => ? = 2
[3,1,3,1,2] => 1001100110 => [1,2,2,2,2,1] => ? = 2
[3,1,3,2,1] => 1001100101 => [1,2,2,2,1,1,1] => ? = 2
[3,2,1,2,2] => 1001011010 => [1,2,1,1,2,1,1,1] => ? = 2
[3,2,1,3,1] => 1001011001 => [1,2,1,1,2,2,1] => ? = 2
[3,2,2,1,2] => 1001010110 => [1,2,1,1,1,1,2,1] => ? = 2
[3,2,2,2,1] => 1001010101 => [1,2,1,1,1,1,1,1,1] => ? = 2
[3,3,1,1,2] => 1001001110 => [1,2,1,2,3,1] => ? = 3
[4,1,1,1,3] => 1000111100 => [1,3,4,2] => ? = 4
[4,1,1,2,2] => 1000111010 => [1,3,3,1,1,1] => ? = 3
[4,1,2,1,2] => 1000110110 => [1,3,2,1,2,1] => ? = 3
Description
The largest part of an integer composition.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 90%
Mp00105: Binary words —complement⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 90%
Values
[1] => 1 => 0 => 0 => 1
[1,1] => 11 => 00 => 01 => 2
[2] => 10 => 01 => 00 => 1
[1,1,1] => 111 => 000 => 010 => 3
[1,2] => 110 => 001 => 011 => 2
[2,1] => 101 => 010 => 000 => 1
[3] => 100 => 011 => 001 => 2
[1,1,1,1] => 1111 => 0000 => 0101 => 4
[1,1,2] => 1110 => 0001 => 0100 => 3
[1,2,1] => 1101 => 0010 => 0111 => 2
[1,3] => 1100 => 0011 => 0110 => 2
[2,1,1] => 1011 => 0100 => 0001 => 2
[2,2] => 1010 => 0101 => 0000 => 1
[3,1] => 1001 => 0110 => 0011 => 2
[4] => 1000 => 0111 => 0010 => 3
[1,1,1,1,1] => 11111 => 00000 => 01010 => 5
[1,1,1,2] => 11110 => 00001 => 01011 => 4
[1,1,2,1] => 11101 => 00010 => 01000 => 3
[1,1,3] => 11100 => 00011 => 01001 => 3
[1,2,1,1] => 11011 => 00100 => 01110 => 2
[1,2,2] => 11010 => 00101 => 01111 => 2
[1,3,1] => 11001 => 00110 => 01100 => 2
[1,4] => 11000 => 00111 => 01101 => 3
[2,1,1,1] => 10111 => 01000 => 00010 => 3
[2,1,2] => 10110 => 01001 => 00011 => 2
[2,2,1] => 10101 => 01010 => 00000 => 1
[2,3] => 10100 => 01011 => 00001 => 2
[3,1,1] => 10011 => 01100 => 00110 => 2
[3,2] => 10010 => 01101 => 00111 => 2
[4,1] => 10001 => 01110 => 00100 => 3
[5] => 10000 => 01111 => 00101 => 4
[1,1,1,1,1,1] => 111111 => 000000 => 010101 => 6
[1,1,1,1,2] => 111110 => 000001 => 010100 => 5
[1,1,1,2,1] => 111101 => 000010 => 010111 => 4
[1,1,1,3] => 111100 => 000011 => 010110 => 4
[1,1,2,1,1] => 111011 => 000100 => 010001 => 3
[1,1,2,2] => 111010 => 000101 => 010000 => 3
[1,1,3,1] => 111001 => 000110 => 010011 => 3
[1,1,4] => 111000 => 000111 => 010010 => 3
[1,2,1,1,1] => 110111 => 001000 => 011101 => 3
[1,2,1,2] => 110110 => 001001 => 011100 => 2
[1,2,2,1] => 110101 => 001010 => 011111 => 2
[1,2,3] => 110100 => 001011 => 011110 => 2
[1,3,1,1] => 110011 => 001100 => 011001 => 2
[1,3,2] => 110010 => 001101 => 011000 => 2
[1,4,1] => 110001 => 001110 => 011011 => 3
[1,5] => 110000 => 001111 => 011010 => 4
[2,1,1,1,1] => 101111 => 010000 => 000101 => 4
[2,1,1,2] => 101110 => 010001 => 000100 => 3
[2,1,2,1] => 101101 => 010010 => 000111 => 2
[1,1,1,1,1,1,1,1,2] => 1111111110 => 0000000001 => 0101010100 => ? = 9
[1,1,1,1,1,1,1,2,1] => 1111111101 => 0000000010 => 0101010111 => ? = 8
[1,1,1,1,6] => 1111100000 => 0000011111 => 0101001010 => ? = 5
[1,1,1,2,5] => 1111010000 => 0000101111 => 0101111010 => ? = 4
[1,1,1,3,4] => 1111001000 => 0000110111 => 0101100010 => ? = 4
[1,1,1,4,3] => 1111000100 => 0000111011 => 0101101110 => ? = 4
[1,1,1,5,2] => 1111000010 => 0000111101 => 0101101000 => ? = 4
[1,1,1,6,1] => 1111000001 => 0000111110 => 0101101011 => ? = 5
[1,1,2,1,1,1,1,2] => 1110111110 => 0001000001 => 0100010100 => ? = 5
[1,1,2,1,5] => 1110110000 => 0001001111 => 0100011010 => ? = 4
[1,1,2,2,4] => 1110101000 => 0001010111 => 0100000010 => ? = 3
[1,1,2,3,3] => 1110100100 => 0001011011 => 0100001110 => ? = 3
[1,1,2,4,2] => 1110100010 => 0001011101 => 0100001000 => ? = 3
[1,1,2,5,1] => 1110100001 => 0001011110 => 0100001011 => ? = 4
[1,1,3,1,1,1,2] => 1110011110 => 0001100001 => 0100110100 => ? = 4
[1,1,3,1,4] => 1110011000 => 0001100111 => 0100110010 => ? = 3
[1,1,3,2,3] => 1110010100 => 0001101011 => 0100111110 => ? = 3
[1,1,3,3,2] => 1110010010 => 0001101101 => 0100111000 => ? = 3
[1,1,3,4,1] => 1110010001 => 0001101110 => 0100111011 => ? = 3
[1,1,4,1,3] => 1110001100 => 0001110011 => 0100100110 => ? = 3
[1,1,4,2,2] => 1110001010 => 0001110101 => 0100100000 => ? = 3
[1,1,4,3,1] => 1110001001 => 0001110110 => 0100100011 => ? = 3
[1,1,5,2,1] => 1110000101 => 0001111010 => 0100101111 => ? = 4
[1,2,1,1,1,1,1,2] => 1101111110 => 0010000001 => 0111010100 => ? = 6
[1,2,1,1,5] => 1101110000 => 0010001111 => 0111011010 => ? = 4
[1,2,1,2,4] => 1101101000 => 0010010111 => 0111000010 => ? = 3
[1,2,1,3,3] => 1101100100 => 0010011011 => 0111001110 => ? = 2
[1,2,1,4,2] => 1101100010 => 0010011101 => 0111001000 => ? = 3
[1,2,1,5,1] => 1101100001 => 0010011110 => 0111001011 => ? = 4
[1,2,2,1,4] => 1101011000 => 0010100111 => 0111110010 => ? = 3
[1,2,2,2,3] => 1101010100 => 0010101011 => 0111111110 => ? = 2
[1,2,2,3,2] => 1101010010 => 0010101101 => 0111111000 => ? = 2
[1,2,2,4,1] => 1101010001 => 0010101110 => 0111111011 => ? = 3
[1,2,3,1,1,2] => 1101001110 => 0010110001 => 0111100100 => ? = 3
[1,2,3,1,3] => 1101001100 => 0010110011 => 0111100110 => ? = 2
[1,2,3,2,2] => 1101001010 => 0010110101 => 0111100000 => ? = 2
[1,2,3,3,1] => 1101001001 => 0010110110 => 0111100011 => ? = 2
[1,2,4,2,1] => 1101000101 => 0010111010 => 0111101111 => ? = 3
[1,3,1,1,4] => 1100111000 => 0011000111 => 0110010010 => ? = 3
[1,3,1,2,3] => 1100110100 => 0011001011 => 0110011110 => ? = 2
[1,3,1,3,2] => 1100110010 => 0011001101 => 0110011000 => ? = 2
[1,3,1,4,1] => 1100110001 => 0011001110 => 0110011011 => ? = 3
[1,3,2,1,3] => 1100101100 => 0011010011 => 0110000110 => ? = 2
[1,3,2,2,2] => 1100101010 => 0011010101 => 0110000000 => ? = 2
[1,3,2,3,1] => 1100101001 => 0011010110 => 0110000011 => ? = 2
[1,3,3,1,2] => 1100100110 => 0011011001 => 0110001100 => ? = 2
[1,3,3,2,1] => 1100100101 => 0011011010 => 0110001111 => ? = 2
[1,4,1,3,1] => 1100011001 => 0011100110 => 0110110011 => ? = 3
[1,4,2,2,1] => 1100010101 => 0011101010 => 0110111111 => ? = 3
[1,4,3,2] => 1100010010 => 0011101101 => 0110111000 => ? = 3
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: St000013
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1] => [1,0]
=> 1
[1,1] => 11 => [2] => [1,1,0,0]
=> 2
[2] => 10 => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => 111 => [3] => [1,1,1,0,0,0]
=> 3
[1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,1,1] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[2,2] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[3,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,1] => 11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,2,1] => 11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,3] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,1,1] => 11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,2,2] => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,3,1] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,1,1] => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[2,1,2] => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[2,2,1] => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[2,3] => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[3,1,1] => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[3,2] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[4,1] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[5] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1,1] => 111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => 111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,2,1] => 111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,1,1,3] => 111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,2,1,1] => 111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,2,2] => 111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[1,1,3,1] => 111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,4] => 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,2,1,1,1] => 110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,2] => 110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,2,1] => 110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,2,3] => 110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,3,1,1] => 110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,2] => 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,1] => 110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[1,5] => 110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1,1] => 101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,2] => 101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[2,1,2,1] => 101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,2,1,1] => 11111011 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,1,1,1,3,1] => 11111001 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[1,1,1,2,1,1,1] => 11110111 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,1,1,4,1] => 11110001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,1,2,1,1,1,1] => 11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,2,1,2,1] => 11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,2,3,1] => 11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,1,3,1,1,1] => 11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,3,2,1] => 11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,4,1,1] => 11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,2,1,1,2,1] => 11011101 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,2,1,2,1,1] => 11011011 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,2,1,2,2] => 11011010 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,2,1,3,1] => 11011001 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,2,3,1,1] => 11010011 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,2,4,1] => 11010001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,3,1,2,1] => 11001101 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,3,2,1,1] => 11001011 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,3,3,1] => 11001001 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,4,1,1,1] => 11000111 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[2,1,1,1,2,1] => 10111101 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[2,1,1,2,1,1] => 10111011 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[2,1,1,3,1] => 10111001 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,1,2,1,1,1] => 10110111 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[2,1,2,1,2] => 10110110 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[2,1,3,2] => 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[2,1,4,1] => 10110001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[2,3,1,1,1] => 10100111 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[2,3,1,2] => 10100110 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[2,4,2] => 10100010 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[2,5,1] => 10100001 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[3,1,1,2,1] => 10011101 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[3,1,2,1,1] => 10011011 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[3,1,2,2] => 10011010 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[3,2,1,1,1] => 10010111 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[3,2,1,2] => 10010110 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[3,3,1,1] => 10010011 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[3,3,2] => 10010010 => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[3,4,1] => 10010001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[4,1,2,1] => 10001101 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[4,2,1,1] => 10001011 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3
[4,3,1] => 10001001 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[5,1,1,1] => 10000111 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4
[5,2,1] => 10000101 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,2,1,1] => 111111011 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,3,1] => 111111001 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[1,1,1,1,2,1,1,1] => 111110111 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,2,1,2] => 111110110 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[1,1,1,1,2,3] => 111110100 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 5
[1,1,1,1,3,1,1] => 111110011 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000444
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 70%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 70%
Values
[1] => 1 => [1] => [1,0]
=> ? = 1
[1,1] => 11 => [2] => [1,1,0,0]
=> 2
[2] => 10 => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => 111 => [3] => [1,1,1,0,0,0]
=> 3
[1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,1,1] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,1] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[2,2] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[3,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,1] => 11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,2,1] => 11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,3] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,1,1] => 11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,2,2] => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,3,1] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,1,1] => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[2,1,2] => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[2,2,1] => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[2,3] => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[3,1,1] => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[3,2] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[4,1] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[5] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1,1] => 111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => 111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,2,1] => 111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,1,1,3] => 111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,2,1,1] => 111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,2,2] => 111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[1,1,3,1] => 111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,1,4] => 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,2,1,1,1] => 110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,2] => 110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,2,1] => 110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,2,3] => 110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,3,1,1] => 110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,2] => 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,1] => 110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[1,5] => 110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1,1] => 101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,2] => 101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[2,1,2,1] => 101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[2,1,3] => 101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,1,1,1,1] => 11111111 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,1,1,1,1,1,2] => 11111110 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,2,1] => 11111101 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,3] => 11111100 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,2,1,1] => 11111011 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,1,1,1,2,2] => 11111010 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,3,1] => 11111001 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
[1,1,1,1,4] => 11111000 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,2,1,1,1] => 11110111 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,1,1,2,1,2] => 11110110 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,1,1,2,2,1] => 11110101 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,2,3] => 11110100 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,1,1,3,1,1] => 11110011 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,1,3,2] => 11110010 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[1,1,1,4,1] => 11110001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,1,1,5] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,2,1,1,1,1] => 11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,2,1,1,2] => 11101110 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[1,1,2,1,2,1] => 11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,2,1,3] => 11101100 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,2,2,1,1] => 11101011 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[1,1,2,2,2] => 11101010 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,1,2,3,1] => 11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,1,2,4] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,1,3,1,1,1] => 11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,3,1,2] => 11100110 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,1,3,2,1] => 11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,3,3] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,1,4,1,1] => 11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,1,4,2] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,5,1] => 11100001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[1,1,6] => 11100000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,1,1,1,1,1,1,1] => 111111111 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => 111111110 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,2,1] => 111111101 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7
[1,1,1,1,1,1,3] => 111111100 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,2,1,1] => 111111011 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,2,2] => 111111010 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,3,1] => 111111001 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,4] => 111111000 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,2,1,1,1] => 111110111 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,2,1,2] => 111110110 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[1,1,1,1,2,2,1] => 111110101 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,2,3] => 111110100 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 5
[1,1,1,1,3,1,1] => 111110011 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[1,1,1,1,3,2] => 111110010 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,1,4,1] => 111110001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[1,1,1,1,5] => 111110000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,1,1,2,1,1,1,1] => 111101111 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 70%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 70%
Values
[1] => 1 => [1] => [1,0]
=> ? = 1 - 1
[1,1] => 11 => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2] => 10 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,1] => 111 => [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2] => 110 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3] => 100 => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1] => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,2] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,1] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,1] => 11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,2] => 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,2,1] => 11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,3] => 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,2,1,1] => 11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,2] => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,1] => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4] => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,1,1] => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,2] => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,2,1] => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,3] => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,1] => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2] => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,1] => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[5] => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,1] => 111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,2] => 111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,2,1] => 111101 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,3] => 111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,2,1,1] => 111011 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,2,2] => 111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,3,1] => 111001 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,4] => 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,1,1,1] => 110111 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,1,2] => 110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,2,1] => 110101 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => 110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,1,1] => 110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2] => 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,1] => 110001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,5] => 110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,1,1,1,1] => 101111 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,1,1,2] => 101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,1,2,1] => 101101 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,3] => 101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,1,1,1] => 11111111 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[1,1,1,1,1,1,2] => 11111110 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,2,1] => 11111101 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,3] => 11111100 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,2,1,1] => 11111011 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,1,2,2] => 11111010 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[1,1,1,1,3,1] => 11111001 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,4] => 11111000 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,1,1,2,1,1,1] => 11110111 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,1,1,2,1,2] => 11110110 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[1,1,1,2,2,1] => 11110101 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,1,1,2,3] => 11110100 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,3,1,1] => 11110011 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,3,2] => 11110010 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,1,1,4,1] => 11110001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
[1,1,1,5] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,2,1,1,1,1] => 11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,1,2,1,1,2] => 11101110 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[1,1,2,1,2,1] => 11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,2,1,3] => 11101100 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,2,2,1,1] => 11101011 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,2,2,2] => 11101010 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,2,3,1] => 11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,2,4] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,3,1,1,1] => 11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,3,1,2] => 11100110 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,1,3,2,1] => 11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,3,3] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,4,1,1] => 11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[1,1,4,2] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,5,1] => 11100001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[1,1,6] => 11100000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[1,1,1,1,1,1,1,1,1] => 111111111 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[1,1,1,1,1,1,1,2] => 111111110 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,2,1] => 111111101 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[1,1,1,1,1,1,3] => 111111100 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,2,1,1] => 111111011 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,1,1,2,2] => 111111010 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,3,1] => 111111001 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 - 1
[1,1,1,1,1,4] => 111111000 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,1,1,1,2,1,1,1] => 111110111 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,1,1,1,2,1,2] => 111110110 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,2,2,1] => 111110101 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[1,1,1,1,2,3] => 111110100 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,1,3,1,1] => 111110011 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 - 1
[1,1,1,1,3,2] => 111110010 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 5 - 1
[1,1,1,1,4,1] => 111110001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 - 1
[1,1,1,1,5] => 111110000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[1,1,1,2,1,1,1,1] => 111101111 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000774
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 70%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 70%
Values
[1] => [1] => ([],1)
=> 1
[1,1] => [2] => ([],2)
=> 2
[2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1] => [3] => ([],3)
=> 3
[1,2] => [1,2] => ([(1,2)],3)
=> 2
[2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,1] => [4] => ([],4)
=> 4
[1,1,2] => [1,3] => ([(2,3)],4)
=> 3
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,1] => [5] => ([],5)
=> 5
[1,1,1,2] => [1,4] => ([(3,4)],5)
=> 4
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1,1] => [6] => ([],6)
=> 6
[1,1,1,1,2] => [1,5] => ([(4,5)],6)
=> 5
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
=> 4
[1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 4
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,1,1,1,1] => [8] => ([],8)
=> ? = 8
[1,1,1,1,1,1,2] => [1,7] => ([(6,7)],8)
=> ? = 7
[1,1,1,1,1,2,1] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,1,3] => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,1,1,2,1,1] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,1,2,2] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,1,3,1] => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,1,4] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,1,2,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,2,1,2] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,2,2,1] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,2,3] => [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,3,1,1] => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,3,2] => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,4,1] => [2,1,1,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,5] => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,2,1,1,1,1] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,1,2,1,1,2] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,2,1,2,1] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,2,1,3] => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,2,2,1,1] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,2,2,2] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,2,3,1] => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,2,4] => [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,3,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,3,1,2] => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,3,2,1] => [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,3,3] => [1,1,2,1,3] => ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,4,1,1] => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,4,2] => [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,5,1] => [2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,6] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,1,1,1,1] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,1,1,2] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,2,1] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,1,1,3] => [1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,1,2,1,1] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,1,2,2] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,1,3,1] => [2,1,3,2] => ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,1,4] => [1,1,1,3,2] => ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,2,1,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,2,1,2] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,2,2,1] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,2,3] => [1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,3,1,1] => [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,3,2] => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,4,1] => [2,1,1,2,2] => ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,5] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,3,1,1,1,1] => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St001652
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 70%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 70%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 2
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 6
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 5
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 5
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? = 4
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => ? = 4
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 4
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,1,2,7,3,4,5] => ? = 3
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => ? = 3
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 3
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => ? = 3
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => ? = 3
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 4
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => ? = 3
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => ? = 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => ? = 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,4] => ? = 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 3
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => ? = 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,7,3,4] => ? = 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 3
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 5
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? = 4
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ? = 3
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ? = 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => ? = 2
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,6,2,3,7,4] => ? = 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 3
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ? = 2
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => ? = 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 2
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 2
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 4
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,2,3,6,7,4] => ? = 3
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,3,6,7,4] => ? = 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,5,1,2,6,7,3] => ? = 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 2
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 3
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => ? = 3
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,2] => ? = 3
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => ? = 4
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 6
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? = 8
[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? = 7
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => ? = 6
[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => ? = 6
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 5
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6] => ? = 5
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => ? = 5
Description
The length of a longest interval of consecutive numbers.
For a permutation $\pi=\pi_1,\dots,\pi_n$, this statistic returns the length of a longest subsequence $\pi_k,\dots,\pi_\ell$ such that $\pi_{i+1} = \pi_i + 1$ for $i\in\{k,\dots,\ell-1\}$.
Matching statistic: St000887
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000887: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 70%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000887: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 70%
Values
[1] => [1,0]
=> [1,0]
=> [1] => ? = 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 2
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,1,2,3,4,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,6,1,2,3,4] => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,1,6,2,3,4] => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,6,2,3,4] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => 3
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 6
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 5
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 5
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? = 4
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => ? = 4
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 4
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,1,2,7,3,4,5] => ? = 3
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => ? = 3
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ? = 3
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => ? = 3
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => ? = 3
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 4
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => ? = 3
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => ? = 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? = 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => ? = 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,4] => ? = 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [4,5,6,1,7,2,3] => ? = 3
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => ? = 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,7,3,4] => ? = 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [4,5,1,6,7,2,3] => ? = 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,1,5,6,7,2,3] => ? = 3
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ? = 5
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? = 4
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ? = 3
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ? = 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => ? = 2
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,5,6,2,3,7,4] => ? = 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ? = 3
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ? = 2
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => ? = 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ? = 2
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ? = 2
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,1,7,2] => ? = 4
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,2,3,6,7,4] => ? = 3
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,3,6,7,4] => ? = 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,5,1,2,6,7,3] => ? = 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ? = 2
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ? = 3
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => ? = 3
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,2] => ? = 3
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => ? = 4
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 6
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? = 8
[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? = 7
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => ? = 6
[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => ? = 6
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 5
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6] => ? = 5
Description
The maximal number of nonzero entries on a diagonal of a permutation matrix.
For example, the permutation matrix of $\pi=[3,1,2,5,4]$ is $$\begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix},$$ and the entries corresponding to $\pi_2=1$, $\pi_3=2$ and $\pi_5=4$ are all on the fourth diagonal from the right.
In other words, this is $\max_k \lvert\{i: \pi_i-i = k\}\rvert$
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!