- FindStat

Your data matches 197 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000383
(load all 26 compositions to match this statistic)

Mapping

St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 2
[2,1] => 1
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 1
[1,3] => 3
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 1
[1,1,3] => 3
[1,2,1,1] => 1
[1,2,2] => 2
[1,3,1] => 1
[1,4] => 4
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 1
[2,3] => 3
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 1
[1,1,1,3] => 3
[1,1,2,1,1] => 1
[1,1,2,2] => 2
[1,1,3,1] => 1
[1,1,4] => 4
[1,2,1,1,1] => 1
[1,2,1,2] => 2
[1,2,2,1] => 1
[1,2,3] => 3
[1,3,1,1] => 1
[1,3,2] => 2
[1,4,1] => 1
[1,5] => 5
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 1

Description

The last part of an integer composition.

Matching statistic: St000382
(load all 16 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1
[1,1] => [1,1] => 1
[2] => [2] => 2
[1,1,1] => [1,1,1] => 1
[1,2] => [2,1] => 2
[2,1] => [1,2] => 1
[3] => [3] => 3
[1,1,1,1] => [1,1,1,1] => 1
[1,1,2] => [2,1,1] => 2
[1,2,1] => [1,2,1] => 1
[1,3] => [3,1] => 3
[2,1,1] => [1,1,2] => 1
[2,2] => [2,2] => 2
[3,1] => [1,3] => 1
[4] => [4] => 4
[1,1,1,1,1] => [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1] => 2
[1,1,2,1] => [1,2,1,1] => 1
[1,1,3] => [3,1,1] => 3
[1,2,1,1] => [1,1,2,1] => 1
[1,2,2] => [2,2,1] => 2
[1,3,1] => [1,3,1] => 1
[1,4] => [4,1] => 4
[2,1,1,1] => [1,1,1,2] => 1
[2,1,2] => [2,1,2] => 2
[2,2,1] => [1,2,2] => 1
[2,3] => [3,2] => 3
[3,1,1] => [1,1,3] => 1
[3,2] => [2,3] => 2
[4,1] => [1,4] => 1
[5] => [5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1] => 2
[1,1,1,2,1] => [1,2,1,1,1] => 1
[1,1,1,3] => [3,1,1,1] => 3
[1,1,2,1,1] => [1,1,2,1,1] => 1
[1,1,2,2] => [2,2,1,1] => 2
[1,1,3,1] => [1,3,1,1] => 1
[1,1,4] => [4,1,1] => 4
[1,2,1,1,1] => [1,1,1,2,1] => 1
[1,2,1,2] => [2,1,2,1] => 2
[1,2,2,1] => [1,2,2,1] => 1
[1,2,3] => [3,2,1] => 3
[1,3,1,1] => [1,1,3,1] => 1
[1,3,2] => [2,3,1] => 2
[1,4,1] => [1,4,1] => 1
[1,5] => [5,1] => 5
[2,1,1,1,1] => [1,1,1,1,2] => 1
[2,1,1,2] => [2,1,1,2] => 2
[2,1,2,1] => [1,2,1,2] => 1

Description

The first part of an integer composition.

Matching statistic: St000326
(load all 9 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 90%

Values

[1] => 1 => 1 => 1
[1,1] => 11 => 11 => 1
[2] => 10 => 01 => 2
[1,1,1] => 111 => 111 => 1
[1,2] => 110 => 011 => 2
[2,1] => 101 => 101 => 1
[3] => 100 => 001 => 3
[1,1,1,1] => 1111 => 1111 => 1
[1,1,2] => 1110 => 0111 => 2
[1,2,1] => 1101 => 1011 => 1
[1,3] => 1100 => 0011 => 3
[2,1,1] => 1011 => 1101 => 1
[2,2] => 1010 => 0101 => 2
[3,1] => 1001 => 1001 => 1
[4] => 1000 => 0001 => 4
[1,1,1,1,1] => 11111 => 11111 => 1
[1,1,1,2] => 11110 => 01111 => 2
[1,1,2,1] => 11101 => 10111 => 1
[1,1,3] => 11100 => 00111 => 3
[1,2,1,1] => 11011 => 11011 => 1
[1,2,2] => 11010 => 01011 => 2
[1,3,1] => 11001 => 10011 => 1
[1,4] => 11000 => 00011 => 4
[2,1,1,1] => 10111 => 11101 => 1
[2,1,2] => 10110 => 01101 => 2
[2,2,1] => 10101 => 10101 => 1
[2,3] => 10100 => 00101 => 3
[3,1,1] => 10011 => 11001 => 1
[3,2] => 10010 => 01001 => 2
[4,1] => 10001 => 10001 => 1
[5] => 10000 => 00001 => 5
[1,1,1,1,1,1] => 111111 => 111111 => 1
[1,1,1,1,2] => 111110 => 011111 => 2
[1,1,1,2,1] => 111101 => 101111 => 1
[1,1,1,3] => 111100 => 001111 => 3
[1,1,2,1,1] => 111011 => 110111 => 1
[1,1,2,2] => 111010 => 010111 => 2
[1,1,3,1] => 111001 => 100111 => 1
[1,1,4] => 111000 => 000111 => 4
[1,2,1,1,1] => 110111 => 111011 => 1
[1,2,1,2] => 110110 => 011011 => 2
[1,2,2,1] => 110101 => 101011 => 1
[1,2,3] => 110100 => 001011 => 3
[1,3,1,1] => 110011 => 110011 => 1
[1,3,2] => 110010 => 010011 => 2
[1,4,1] => 110001 => 100011 => 1
[1,5] => 110000 => 000011 => 5
[2,1,1,1,1] => 101111 => 111101 => 1
[2,1,1,2] => 101110 => 011101 => 2
[2,1,2,1] => 101101 => 101101 => 1
[1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
[1,1,1,1,2,1,1,2] => 1111101110 => 0111011111 => ? = 2
[1,1,1,1,2,2,1,1] => 1111101011 => 1101011111 => ? = 1
[1,1,1,1,5,1] => 1111100001 => 1000011111 => ? = 1
[1,1,2,1,1,1,1,2] => 1110111110 => 0111110111 => ? = 2
[1,1,2,1,1,2,1,1] => 1110111011 => 1101110111 => ? = 1
[1,1,2,1,2,3] => 1110110100 => 0010110111 => ? = 3
[1,1,2,2,1,1,1,1] => 1110101111 => 1111010111 => ? = 1
[1,1,2,2,2,2] => 1110101010 => 0101010111 => ? = 2
[1,1,3,1,1,3] => 1110011100 => 0011100111 => ? = 3
[1,1,3,2,1,2] => 1110010110 => 0110100111 => ? = 2
[1,1,3,3,1,1] => 1110010011 => 1100100111 => ? = 1
[1,1,7,1] => 1110000001 => 1000000111 => ? = 1
[1,2,1,1,4,1] => 1101110001 => 1000111011 => ? = 1
[1,2,6,1] => 1101000001 => 1000001011 => ? = 1
[1,3,1,1,3,1] => 1100111001 => 1001110011 => ? = 1
[1,3,5,1] => 1100100001 => 1000010011 => ? = 1
[1,4,1,1,2,1] => 1100011101 => 1011100011 => ? = 1
[1,4,4,1] => 1100010001 => 1000100011 => ? = 1
[1,5,1,1,1,1] => 1100001111 => 1111000011 => ? = 1
[1,5,3,1] => 1100001001 => 1001000011 => ? = 1
[1,6,1,2] => 1100000110 => 0110000011 => ? = 2
[1,6,2,1] => 1100000101 => 1010000011 => ? = 1
[1,7,1,1] => 1100000011 => 1100000011 => ? = 1
[1,8,1] => 1100000001 => 1000000011 => ? = 1
[2,1,1,1,1,2,1,1] => 1011111011 => 1101111101 => ? = 1
[2,1,1,2,1,1,1,1] => 1011101111 => 1111011101 => ? = 1
[2,1,1,2,2,2] => 1011101010 => 0101011101 => ? = 2
[2,1,2,1,1,3] => 1011011100 => 0011101101 => ? = 3
[2,1,2,2,1,2] => 1011010110 => 0110101101 => ? = 2
[2,1,2,3,1,1] => 1011010011 => 1100101101 => ? = 1
[2,1,6,1] => 1011000001 => 1000001101 => ? = 1
[2,2,1,1,1,1,1,1] => 1010111111 => 1111110101 => ? = 1
[2,2,1,1,2,2] => 1010111010 => 0101110101 => ? = 2
[2,2,2,1,1,2] => 1010101110 => 0111010101 => ? = 2
[2,2,2,2,1,1] => 1010101011 => 1101010101 => ? = 1
[3,1,1,1,1,3] => 1001111100 => 0011111001 => ? = 3
[3,1,1,2,1,2] => 1001110110 => 0110111001 => ? = 2
[3,1,1,3,1,1] => 1001110011 => 1100111001 => ? = 1
[3,2,1,2,1,1] => 1001011011 => 1101101001 => ? = 1
[3,3,1,1,1,1] => 1001001111 => 1111001001 => ? = 1
[3,3,2,2] => 1001001010 => 0101001001 => ? = 2
[4,2,1,3] => 1000101100 => 0011010001 => ? = 3
[4,3,1,2] => 1000100110 => 0110010001 => ? = 2
[4,4,1,1] => 1000100011 => 1100010001 => ? = 1
[8,1,1] => 1000000011 => 1100000001 => ? = 1
[9,1] => 1000000001 => 1000000001 => ? = 1
[1,10] => 11000000000 => 00000000011 => ? = 10
[1,8,1,1] => 11000000011 => 11000000011 => ? = 1
[1,7,2,1] => 11000000101 => 10100000011 => ? = 1

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St000297
(load all 9 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 80%

Values

[1] => 1 => 1 => 0 => 0 = 1 - 1
[1,1] => 11 => 11 => 00 => 0 = 1 - 1
[2] => 10 => 01 => 10 => 1 = 2 - 1
[1,1,1] => 111 => 111 => 000 => 0 = 1 - 1
[1,2] => 110 => 011 => 100 => 1 = 2 - 1
[2,1] => 101 => 101 => 010 => 0 = 1 - 1
[3] => 100 => 001 => 110 => 2 = 3 - 1
[1,1,1,1] => 1111 => 1111 => 0000 => 0 = 1 - 1
[1,1,2] => 1110 => 0111 => 1000 => 1 = 2 - 1
[1,2,1] => 1101 => 1011 => 0100 => 0 = 1 - 1
[1,3] => 1100 => 0011 => 1100 => 2 = 3 - 1
[2,1,1] => 1011 => 1101 => 0010 => 0 = 1 - 1
[2,2] => 1010 => 0101 => 1010 => 1 = 2 - 1
[3,1] => 1001 => 1001 => 0110 => 0 = 1 - 1
[4] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,1,1,1,1] => 11111 => 11111 => 00000 => 0 = 1 - 1
[1,1,1,2] => 11110 => 01111 => 10000 => 1 = 2 - 1
[1,1,2,1] => 11101 => 10111 => 01000 => 0 = 1 - 1
[1,1,3] => 11100 => 00111 => 11000 => 2 = 3 - 1
[1,2,1,1] => 11011 => 11011 => 00100 => 0 = 1 - 1
[1,2,2] => 11010 => 01011 => 10100 => 1 = 2 - 1
[1,3,1] => 11001 => 10011 => 01100 => 0 = 1 - 1
[1,4] => 11000 => 00011 => 11100 => 3 = 4 - 1
[2,1,1,1] => 10111 => 11101 => 00010 => 0 = 1 - 1
[2,1,2] => 10110 => 01101 => 10010 => 1 = 2 - 1
[2,2,1] => 10101 => 10101 => 01010 => 0 = 1 - 1
[2,3] => 10100 => 00101 => 11010 => 2 = 3 - 1
[3,1,1] => 10011 => 11001 => 00110 => 0 = 1 - 1
[3,2] => 10010 => 01001 => 10110 => 1 = 2 - 1
[4,1] => 10001 => 10001 => 01110 => 0 = 1 - 1
[5] => 10000 => 00001 => 11110 => 4 = 5 - 1
[1,1,1,1,1,1] => 111111 => 111111 => 000000 => 0 = 1 - 1
[1,1,1,1,2] => 111110 => 011111 => 100000 => 1 = 2 - 1
[1,1,1,2,1] => 111101 => 101111 => 010000 => 0 = 1 - 1
[1,1,1,3] => 111100 => 001111 => 110000 => 2 = 3 - 1
[1,1,2,1,1] => 111011 => 110111 => 001000 => 0 = 1 - 1
[1,1,2,2] => 111010 => 010111 => 101000 => 1 = 2 - 1
[1,1,3,1] => 111001 => 100111 => 011000 => 0 = 1 - 1
[1,1,4] => 111000 => 000111 => 111000 => 3 = 4 - 1
[1,2,1,1,1] => 110111 => 111011 => 000100 => 0 = 1 - 1
[1,2,1,2] => 110110 => 011011 => 100100 => 1 = 2 - 1
[1,2,2,1] => 110101 => 101011 => 010100 => 0 = 1 - 1
[1,2,3] => 110100 => 001011 => 110100 => 2 = 3 - 1
[1,3,1,1] => 110011 => 110011 => 001100 => 0 = 1 - 1
[1,3,2] => 110010 => 010011 => 101100 => 1 = 2 - 1
[1,4,1] => 110001 => 100011 => 011100 => 0 = 1 - 1
[1,5] => 110000 => 000011 => 111100 => 4 = 5 - 1
[2,1,1,1,1] => 101111 => 111101 => 000010 => 0 = 1 - 1
[2,1,1,2] => 101110 => 011101 => 100010 => 1 = 2 - 1
[2,1,2,1] => 101101 => 101101 => 010010 => 0 = 1 - 1
[1,1,1,1,1,1,2,2] => 1111111010 => 0101111111 => 1010000000 => ? = 2 - 1
[1,1,1,1,2,1,1,2] => 1111101110 => 0111011111 => 1000100000 => ? = 2 - 1
[1,1,1,1,3,3] => 1111100100 => 0010011111 => 1101100000 => ? = 3 - 1
[1,1,1,1,5,1] => 1111100001 => 1000011111 => 0111100000 => ? = 1 - 1
[1,1,2,1,1,1,1,2] => 1110111110 => 0111110111 => 1000001000 => ? = 2 - 1
[1,1,2,1,2,3] => 1110110100 => 0010110111 => 1101001000 => ? = 3 - 1
[1,1,2,2,2,2] => 1110101010 => 0101010111 => 1010101000 => ? = 2 - 1
[1,1,3,1,1,3] => 1110011100 => 0011100111 => 1100011000 => ? = 3 - 1
[1,1,3,2,1,2] => 1110010110 => 0110100111 => 1001011000 => ? = 2 - 1
[1,1,3,3,1,1] => 1110010011 => 1100100111 => 0011011000 => ? = 1 - 1
[1,1,4,4] => 1110001000 => 0001000111 => 1110111000 => ? = 4 - 1
[1,1,7,1] => 1110000001 => 1000000111 => 0111111000 => ? = 1 - 1
[1,2,1,1,4,1] => 1101110001 => 1000111011 => 0111000100 => ? = 1 - 1
[1,2,6,1] => 1101000001 => 1000001011 => 0111110100 => ? = 1 - 1
[1,3,1,1,3,1] => 1100111001 => 1001110011 => 0110001100 => ? = 1 - 1
[1,3,5,1] => 1100100001 => 1000010011 => 0111101100 => ? = 1 - 1
[1,4,1,1,2,1] => 1100011101 => 1011100011 => 0100011100 => ? = 1 - 1
[1,4,4,1] => 1100010001 => 1000100011 => 0111011100 => ? = 1 - 1
[1,5,3,1] => 1100001001 => 1001000011 => 0110111100 => ? = 1 - 1
[1,6,1,2] => 1100000110 => 0110000011 => 1001111100 => ? = 2 - 1
[1,6,2,1] => 1100000101 => 1010000011 => 0101111100 => ? = 1 - 1
[1,7,1,1] => 1100000011 => 1100000011 => 0011111100 => ? = 1 - 1
[1,8,1] => 1100000001 => 1000000011 => 0111111100 => ? = 1 - 1
[1,9] => 1100000000 => 0000000011 => 1111111100 => ? = 9 - 1
[2,1,1,1,2,3] => 1011110100 => 0010111101 => 1101000010 => ? = 3 - 1
[2,1,1,2,2,2] => 1011101010 => 0101011101 => 1010100010 => ? = 2 - 1
[2,1,2,2,1,2] => 1011010110 => 0110101101 => 1001010010 => ? = 2 - 1
[2,1,2,3,1,1] => 1011010011 => 1100101101 => 0011010010 => ? = 1 - 1
[2,1,3,4] => 1011001000 => 0001001101 => 1110110010 => ? = 4 - 1
[2,1,6,1] => 1011000001 => 1000001101 => 0111110010 => ? = 1 - 1
[2,2,1,1,2,2] => 1010111010 => 0101110101 => 1010001010 => ? = 2 - 1
[2,2,2,1,1,2] => 1010101110 => 0111010101 => 1000101010 => ? = 2 - 1
[2,2,3,3] => 1010100100 => 0010010101 => 1101101010 => ? = 3 - 1
[3,1,1,1,1,3] => 1001111100 => 0011111001 => 1100000110 => ? = 3 - 1
[3,1,1,2,1,2] => 1001110110 => 0110111001 => 1001000110 => ? = 2 - 1
[3,1,1,3,1,1] => 1001110011 => 1100111001 => 0011000110 => ? = 1 - 1
[3,1,2,4] => 1001101000 => 0001011001 => 1110100110 => ? = 4 - 1
[3,2,2,3] => 1001010100 => 0010101001 => 1101010110 => ? = 3 - 1
[3,3,2,2] => 1001001010 => 0101001001 => 1010110110 => ? = 2 - 1
[4,1,1,4] => 1000111000 => 0001110001 => 1110001110 => ? = 4 - 1
[4,2,1,3] => 1000101100 => 0011010001 => 1100101110 => ? = 3 - 1
[4,3,1,2] => 1000100110 => 0110010001 => 1001101110 => ? = 2 - 1
[4,4,1,1] => 1000100011 => 1100010001 => 0011101110 => ? = 1 - 1
[5,5] => 1000010000 => 0000100001 => 1111011110 => ? = 5 - 1
[8,1,1] => 1000000011 => 1100000001 => 0011111110 => ? = 1 - 1
[9,1] => 1000000001 => 1000000001 => 0111111110 => ? = 1 - 1
[1,10] => 11000000000 => 00000000011 => 11111111100 => ? = 10 - 1
[1,8,1,1] => 11000000011 => 11000000011 => 00111111100 => ? = 1 - 1
[1,7,2,1] => 11000000101 => 10100000011 => 01011111100 => ? = 1 - 1
[1,6,3,1] => 11000001001 => 10010000011 => 01101111100 => ? = 1 - 1

Description

The number of leading ones in a binary word.

Matching statistic: St000645

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1] => [1,0]
 => 0 = 1 - 1
[1,1] => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[2] => [1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2 = 3 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 2 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 2 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [7,8,4,5,6,3,2,1] => ?
 => ? = 2 - 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,2,3,1] => ?
 => ? = 2 - 1
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,2,3,1] => ?
 => ? = 2 - 1
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,2,3,4,1] => ?
 => ? = 2 - 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [8,5,6,7,4,3,1,2] => ?
 => ? = 1 - 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [8,7,4,5,6,3,1,2] => ?
 => ? = 1 - 1
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [8,4,5,6,7,3,1,2] => ?
 => ? = 1 - 1
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,3,4,1,2] => ?
 => ? = 1 - 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,1,2,3] => ?
 => ? = 2 - 1
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [9,5,6,7,8,4,3,2,1] => ?
 => ? = 1 - 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [9,4,5,6,7,8,2,3,1] => ?
 => ? = 1 - 1
[1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,2,3,4,5,1] => ?
 => ? = 1 - 1
[1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [7,8,9,6,2,3,4,5,1] => ?
 => ? = 3 - 1
[1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? => ?
 => ? = 2 - 1
[1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2 - 1
[1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [9,4,5,6,7,8,3,1,2] => ?
 => ? = 1 - 1
[2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [9,5,6,7,8,4,1,2,3] => ?
 => ? = 1 - 1
[1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2 - 1
[1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? => ?
 => ? = 3 - 1
[1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [10,5,6,7,8,9,4,3,2,1] => ?
 => ? = 1 - 1
[1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2 - 1
[1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? => ?
 => ? = 3 - 1
[1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [10,9,8,7,5,6,3,4,2,1] => ?
 => ? = 1 - 1
[1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? => ?
 => ? = 3 - 1
[1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2 - 1
[1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [7,8,9,10,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 4 - 1
[1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,2,6,1] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,3,5,1] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,4,4,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,5,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [10,9,8,7,2,3,4,5,6,1] => ?
 => ? = 1 - 1
[1,5,3,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1 - 1
[1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2 - 1

Description

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. For a Dyck path

$D = D_1 \cdots D_{2n}$ with peaks in positions

$i_1 < \ldots < i_k$ and valleys in positions

$j_1 < \ldots < j_{k-1}$ , this statistic is given by

$\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)$

Matching statistic: St000617
(load all 8 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1] => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 1
[2] => [1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 2
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [7,8,4,5,6,3,2,1] => ?
 => ? = 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,2,3,1] => ?
 => ? = 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,2,3,1] => ?
 => ? = 2
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,2,3,4,1] => ?
 => ? = 2
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [8,5,6,7,4,3,1,2] => ?
 => ? = 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [8,7,4,5,6,3,1,2] => ?
 => ? = 1
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [8,4,5,6,7,3,1,2] => ?
 => ? = 1
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,3,4,1,2] => ?
 => ? = 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,1,2,3] => ?
 => ? = 2
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [9,5,6,7,8,4,3,2,1] => ?
 => ? = 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1
[1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1
[1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [9,4,5,6,7,8,2,3,1] => ?
 => ? = 1
[1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1
[1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? => ?
 => ? = 1
[1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? => ?
 => ? = 1
[1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,2,3,4,5,1] => ?
 => ? = 1
[1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [7,8,9,6,2,3,4,5,1] => ?
 => ? = 3
[1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? => ?
 => ? = 2
[1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2
[1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1
[2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [9,4,5,6,7,8,3,1,2] => ?
 => ? = 1
[2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1
[3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [9,5,6,7,8,4,1,2,3] => ?
 => ? = 1
[1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2
[1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? => ?
 => ? = 1
[1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? => ?
 => ? = 3
[1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [10,5,6,7,8,9,4,3,2,1] => ?
 => ? = 1
[1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2
[1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? => ?
 => ? = 1
[1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? => ?
 => ? = 3
[1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [10,9,8,7,5,6,3,4,2,1] => ?
 => ? = 1
[1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? => ?
 => ? = 3
[1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? => ?
 => ? = 2
[1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? => ?
 => ? = 1
[1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [7,8,9,10,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => ? = 4
[1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,2,6,1] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,3,5,1] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? => ?
 => ? = 1
[1,4,4,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ?
 => ? = 1
[1,5,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [10,9,8,7,2,3,4,5,6,1] => ?
 => ? = 1
[1,5,3,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? => ?
 => ? = 1

Description

The number of global maxima of a Dyck path.

Matching statistic: St000757

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 80%

Values

[1] => [1,0]
 => [1,0]
 => [1] => 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [2] => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1] => 2
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => 1
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [3] => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 4
[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]
 => [5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 5
[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]
 => [6] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [6] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,5] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,5] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? => ? = 1
[1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? => ? = 1
[1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10] => ? = 1
[1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => ? = 2
[1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? => ? = 3
[1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [10] => ? = 1
[1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => ? => ? = 1
[1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,9] => ? = 2
[1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? => ? = 4
[1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [10] => ? = 1
[1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,2,6,1] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,5,1] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? => ? = 1
[1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? => ? = 1
[1,4,4,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? => ? = 1
[1,5,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [10] => ? = 1
[1,5,3,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? => ? = 1
[1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,9] => ? = 2
[1,6,2,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ?
 => ? => ? = 1
[1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [10] => ? = 1
[1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [10] => ? = 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,2] => ? = 9
[2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9] => ? = 2
[2,1,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10] => ? = 1
[2,1,1,1,2,3] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[2,1,1,2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ?
 => ? => ? = 1
[2,1,1,2,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? => ? = 2
[2,1,2,1,1,3] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[2,1,2,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2

Description

The length of the longest weakly inreasing subsequence of parts of an integer composition.

Matching statistic: St000765

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 80%

Values

[1] => [1,0]
 => [1,0]
 => [1] => 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [2] => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1] => 2
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3] => 1
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [3] => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [4] => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 4
[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]
 => [5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 5
[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]
 => [6] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [6] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,5] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,5] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? => ? = 1
[1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? => ? = 1
[1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10] => ? = 1
[1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,9] => ? = 2
[1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? => ? = 3
[1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [10] => ? = 1
[1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => ? => ? = 1
[1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,9] => ? = 2
[1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2
[1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ?
 => ? => ? = 1
[1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => ? => ? = 4
[1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [10] => ? = 1
[1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,2,6,1] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? => ? = 1
[1,3,5,1] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? => ? = 1
[1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? => ? = 1
[1,4,4,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? => ? = 1
[1,5,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [10] => ? = 1
[1,5,3,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? => ? = 1
[1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,9] => ? = 2
[1,6,2,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ?
 => ? => ? = 1
[1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => [10] => ? = 1
[1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [10] => ? = 1
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,2] => ? = 9
[2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9] => ? = 2
[2,1,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [10] => ? = 1
[2,1,1,1,2,3] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[2,1,1,2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ?
 => ? => ? = 1
[2,1,1,2,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? => ? = 2
[2,1,2,1,1,3] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ? => ? = 3
[2,1,2,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ? => ? = 2

Description

The number of weak records in an integer composition. A weak record is an element

$a_i$ such that

$a_i \geq a_j$ for all

$j < i$ .

Matching statistic: St000439
(load all 10 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 2 = 1 + 1
[1,1] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[2] => [2] => [1,1,0,0]
 => 3 = 2 + 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2] => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[2,1] => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3] => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4] => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 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]
 => 2 = 1 + 1
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 5 + 1
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
[1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
[2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
[3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[1,1,1,1,4,1] => [1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,1,1,2,3,1] => [1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,1,2,1,3,1] => [1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,2,1,1,3,1] => [1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,1,2,2,1] => [1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,2,1,2,1] => [1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,5,1] => [1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,3,1,1,2,1] => [1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,3,1,2,1,1] => [1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,3,2,1,1,1] => [1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,3,4,1] => [1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,4,1,1,1,1] => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
[1,4,1,3] => [3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[1,4,2,2] => [2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2 + 1
[1,4,3,1] => [1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
[1,5,1,2] => [2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 + 1
[1,5,2,1] => [1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
[2,1,5,1] => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
[2,2,4,1] => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[3,1,4,1] => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,1,1,1,2,1,1,2] => [2,1,1,2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,1,1,1,2,2,1,1] => [1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,1,1,1,3,3] => [3,3,1,1,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,1,1,1,5,1] => [1,5,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,1,2,1,1,1,1,2] => [2,1,1,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,2,1,1,2,1,1] => [1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,1,2,1,2,3] => [3,2,1,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[1,1,2,2,1,1,1,1] => [1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,1,2,2,2,2] => [2,2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,3,1,1,3] => [3,1,1,3,1,1] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,1,3,2,1,2] => [2,1,2,3,1,1] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,3,3,1,1] => [1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 + 1
[1,1,4,4] => [4,4,1,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[1,2,1,1,4,1] => [1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,6,1] => [1,6,2,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,3,1,1,3,1] => [1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,3,5,1] => [1,5,3,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
[1,4,1,1,2,1] => [1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
[1,4,4,1] => [1,4,4,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1

Description

The position of the first down step of a Dyck path.

Matching statistic: St000678
(load all 13 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 70%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => ? = 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? = 1
[1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => ?
 => ? = 1
[1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ?
 => ? = 1
[1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ?
 => ?
 => ? = 1
[1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ?
 => ? = 3
[1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ?
 => ? = 2
[1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
[2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 1
[2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
[6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ?
 => ?
 => ? = 2
[1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ?
 => ?
 => ? = 1
[1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? = 3
[1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ?
 => ? = 2
[1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ?
 => ? = 1
[1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ?
 => ?
 => ? = 3
[1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 1
[1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ?
 => ?
 => ? = 3
[1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ?
 => ? = 2
[1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ?
 => ?
 => ? = 1
[1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ?
 => ? = 4
[1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,2,6,1] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ?
 => ? = 1
[1,3,5,1] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? = 1
[1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ?
 => ? = 1

Description

The number of up steps after the last double rise of a Dyck path.

The following 187 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000011The number of touch points (or returns) of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000010The length of the partition. St001176The size of a partition minus its first part. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000532The total number of rook placements on a Ferrers board. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St000012The area of a Dyck path. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000984The number of boxes below precisely one peak. St000759The smallest missing part in an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000069The number of maximal elements of a poset. St000068The number of minimal elements in a poset. St000505The biggest entry in the block containing the 1. St000700The protection number of an ordered tree. St000971The smallest closer of a set partition. St000993The multiplicity of the largest part of an integer partition. St001050The number of terminal closers of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001691The number of kings in a graph. St001829The common independence number of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000722The number of different neighbourhoods in a graph. St000908The length of the shortest maximal antichain in a poset. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000287The number of connected components of a graph. St000363The number of minimal vertex covers of a graph. St000553The number of blocks of a graph. St000917The open packing number of a graph. St001672The restrained domination number of a graph. St000286The number of connected components of the complement of a graph. St000822The Hadwiger number of the graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001963The tree-depth of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000237The number of small exceedances. St000007The number of saliances of the permutation. St001812The biclique partition number of a graph. St001828The Euler characteristic of a graph. St000054The first entry of the permutation. St000914The sum of the values of the Möbius function of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000546The number of global descents of a permutation. St001330The hat guessing number of a graph. St000883The number of longest increasing subsequences of a permutation. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001933The largest multiplicity of a part in an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000501The size of the first part in the decomposition of a permutation. St000989The number of final rises of a permutation. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000740The last entry of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000031The number of cycles in the cycle decomposition of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000843The decomposition number of a perfect matching. St000203The number of external nodes of a binary tree. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000090The variation of a composition. St000258The burning number of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama algebra. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000699The toughness times the least common multiple of 1,. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001340The cardinality of a minimal non-edge isolating set of a graph. St000061The number of nodes on the left branch of a binary tree. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000918The 2-limited packing number of a graph. St001480The number of simple summands of the module J^2/J^3. St001462The number of factors of a standard tableaux under concatenation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St001530The depth of a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau.