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Your data matches 26 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147

Mapping

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => 1 => [1,1] => [1,1]
 => 1
1 => 0 => [2] => [2]
 => 2
00 => 11 => [1,1,1] => [1,1,1]
 => 1
01 => 10 => [1,2] => [2,1]
 => 2
10 => 01 => [2,1] => [2,1]
 => 2
11 => 00 => [3] => [3]
 => 3
000 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1
001 => 110 => [1,1,2] => [2,1,1]
 => 2
010 => 101 => [1,2,1] => [2,1,1]
 => 2
011 => 100 => [1,3] => [3,1]
 => 3
100 => 011 => [2,1,1] => [2,1,1]
 => 2
101 => 010 => [2,2] => [2,2]
 => 2
110 => 001 => [3,1] => [3,1]
 => 3
111 => 000 => [4] => [4]
 => 4
0000 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
0001 => 1110 => [1,1,1,2] => [2,1,1,1]
 => 2
0010 => 1101 => [1,1,2,1] => [2,1,1,1]
 => 2
0011 => 1100 => [1,1,3] => [3,1,1]
 => 3
0100 => 1011 => [1,2,1,1] => [2,1,1,1]
 => 2
0101 => 1010 => [1,2,2] => [2,2,1]
 => 2
0110 => 1001 => [1,3,1] => [3,1,1]
 => 3
0111 => 1000 => [1,4] => [4,1]
 => 4
1000 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 2
1001 => 0110 => [2,1,2] => [2,2,1]
 => 2
1010 => 0101 => [2,2,1] => [2,2,1]
 => 2
1011 => 0100 => [2,3] => [3,2]
 => 3
1100 => 0011 => [3,1,1] => [3,1,1]
 => 3
1101 => 0010 => [3,2] => [3,2]
 => 3
1110 => 0001 => [4,1] => [4,1]
 => 4
1111 => 0000 => [5] => [5]
 => 5
00000 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1
00001 => 11110 => [1,1,1,1,2] => [2,1,1,1,1]
 => 2
00010 => 11101 => [1,1,1,2,1] => [2,1,1,1,1]
 => 2
00011 => 11100 => [1,1,1,3] => [3,1,1,1]
 => 3
00100 => 11011 => [1,1,2,1,1] => [2,1,1,1,1]
 => 2
00101 => 11010 => [1,1,2,2] => [2,2,1,1]
 => 2
00110 => 11001 => [1,1,3,1] => [3,1,1,1]
 => 3
00111 => 11000 => [1,1,4] => [4,1,1]
 => 4
01000 => 10111 => [1,2,1,1,1] => [2,1,1,1,1]
 => 2
01001 => 10110 => [1,2,1,2] => [2,2,1,1]
 => 2
01010 => 10101 => [1,2,2,1] => [2,2,1,1]
 => 2
01011 => 10100 => [1,2,3] => [3,2,1]
 => 3
01100 => 10011 => [1,3,1,1] => [3,1,1,1]
 => 3
01101 => 10010 => [1,3,2] => [3,2,1]
 => 3
01110 => 10001 => [1,4,1] => [4,1,1]
 => 4
01111 => 10000 => [1,5] => [5,1]
 => 5
10000 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 2
10001 => 01110 => [2,1,1,2] => [2,2,1,1]
 => 2
10010 => 01101 => [2,1,2,1] => [2,2,1,1]
 => 2
10011 => 01100 => [2,1,3] => [3,2,1]
 => 3

Description

The largest part of an integer partition.

Matching statistic: St000392
(load all 2 compositions to match this statistic)

Mapping

St000392: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 92%

Values

0 => 0 = 1 - 1
1 => 1 = 2 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 2 = 3 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 1
0000 => 0 = 1 - 1
0001 => 1 = 2 - 1
0010 => 1 = 2 - 1
0011 => 2 = 3 - 1
0100 => 1 = 2 - 1
0101 => 1 = 2 - 1
0110 => 2 = 3 - 1
0111 => 3 = 4 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 2 = 3 - 1
1100 => 2 = 3 - 1
1101 => 2 = 3 - 1
1110 => 3 = 4 - 1
1111 => 4 = 5 - 1
00000 => 0 = 1 - 1
00001 => 1 = 2 - 1
00010 => 1 = 2 - 1
00011 => 2 = 3 - 1
00100 => 1 = 2 - 1
00101 => 1 = 2 - 1
00110 => 2 = 3 - 1
00111 => 3 = 4 - 1
01000 => 1 = 2 - 1
01001 => 1 = 2 - 1
01010 => 1 = 2 - 1
01011 => 2 = 3 - 1
01100 => 2 = 3 - 1
01101 => 2 = 3 - 1
01110 => 3 = 4 - 1
01111 => 4 = 5 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 2 = 3 - 1
1111111111 => ? = 11 - 1
1100000001 => ? = 3 - 1
1011111111 => ? = 9 - 1
1010000001 => ? = 2 - 1
1000000101 => ? = 2 - 1
1000000011 => ? = 3 - 1
10000101010 => ? = 2 - 1
10111011110 => ? = 5 - 1
11110111110 => ? = 6 - 1
10111101110 => ? = 5 - 1
10001000000 => ? = 2 - 1
11101011110 => ? = 5 - 1
11011101110 => ? = 4 - 1
10101010110 => ? = 3 - 1
11101101110 => ? = 4 - 1
10100000000 => ? = 2 - 1
10101011010 => ? = 3 - 1
11101110110 => ? = 4 - 1
11111111010 => ? = 9 - 1
 => ? = 1 - 1
11010101010 => ? = 3 - 1
11010101000 => ? = 3 - 1
11010101100 => ? = 3 - 1
11010100100 => ? = 3 - 1
11010010100 => ? = 3 - 1
11011010100 => ? = 3 - 1
11001010100 => ? = 3 - 1
10101010100 => ? = 2 - 1
0110000000 => ? = 3 - 1
0110000110 => ? = 3 - 1
0110011000 => ? = 3 - 1
0110011110 => ? = 5 - 1
0110010010 => ? = 3 - 1
0111100000 => ? = 5 - 1
0111100110 => ? = 5 - 1
0111111000 => ? = 7 - 1
0111111110 => ? = 9 - 1
0111110010 => ? = 6 - 1
0111001000 => ? = 4 - 1
0111000010 => ? = 4 - 1
0100100000 => ? = 2 - 1
0100100110 => ? = 3 - 1
0100111000 => ? = 4 - 1
0100111110 => ? = 6 - 1
0100110010 => ? = 3 - 1
0100001000 => ? = 2 - 1
0100001110 => ? = 4 - 1
0100011010 => ? = 3 - 1
0101101000 => ? = 3 - 1
0101100010 => ? = 3 - 1

Description

The length of the longest run of ones in a binary word.

Matching statistic: St000381
(load all 10 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 83%

Values

0 => 1 => [1,1] => 1
1 => 0 => [2] => 2
00 => 11 => [1,1,1] => 1
01 => 10 => [1,2] => 2
10 => 01 => [2,1] => 2
11 => 00 => [3] => 3
000 => 111 => [1,1,1,1] => 1
001 => 110 => [1,1,2] => 2
010 => 101 => [1,2,1] => 2
011 => 100 => [1,3] => 3
100 => 011 => [2,1,1] => 2
101 => 010 => [2,2] => 2
110 => 001 => [3,1] => 3
111 => 000 => [4] => 4
0000 => 1111 => [1,1,1,1,1] => 1
0001 => 1110 => [1,1,1,2] => 2
0010 => 1101 => [1,1,2,1] => 2
0011 => 1100 => [1,1,3] => 3
0100 => 1011 => [1,2,1,1] => 2
0101 => 1010 => [1,2,2] => 2
0110 => 1001 => [1,3,1] => 3
0111 => 1000 => [1,4] => 4
1000 => 0111 => [2,1,1,1] => 2
1001 => 0110 => [2,1,2] => 2
1010 => 0101 => [2,2,1] => 2
1011 => 0100 => [2,3] => 3
1100 => 0011 => [3,1,1] => 3
1101 => 0010 => [3,2] => 3
1110 => 0001 => [4,1] => 4
1111 => 0000 => [5] => 5
00000 => 11111 => [1,1,1,1,1,1] => 1
00001 => 11110 => [1,1,1,1,2] => 2
00010 => 11101 => [1,1,1,2,1] => 2
00011 => 11100 => [1,1,1,3] => 3
00100 => 11011 => [1,1,2,1,1] => 2
00101 => 11010 => [1,1,2,2] => 2
00110 => 11001 => [1,1,3,1] => 3
00111 => 11000 => [1,1,4] => 4
01000 => 10111 => [1,2,1,1,1] => 2
01001 => 10110 => [1,2,1,2] => 2
01010 => 10101 => [1,2,2,1] => 2
01011 => 10100 => [1,2,3] => 3
01100 => 10011 => [1,3,1,1] => 3
01101 => 10010 => [1,3,2] => 3
01110 => 10001 => [1,4,1] => 4
01111 => 10000 => [1,5] => 5
10000 => 01111 => [2,1,1,1,1] => 2
10001 => 01110 => [2,1,1,2] => 2
10010 => 01101 => [2,1,2,1] => 2
10011 => 01100 => [2,1,3] => 3
000000001 => 111111110 => [1,1,1,1,1,1,1,1,2] => ? = 2
000000010 => 111111101 => [1,1,1,1,1,1,1,2,1] => ? = 2
000000011 => 111111100 => [1,1,1,1,1,1,1,3] => ? = 3
000000100 => 111111011 => [1,1,1,1,1,1,2,1,1] => ? = 2
000000110 => 111111001 => [1,1,1,1,1,1,3,1] => ? = 3
000000111 => 111111000 => [1,1,1,1,1,1,4] => ? = 4
000001000 => 111110111 => [1,1,1,1,1,2,1,1,1] => ? = 2
000001001 => 111110110 => [1,1,1,1,1,2,1,2] => ? = 2
000001010 => 111110101 => [1,1,1,1,1,2,2,1] => ? = 2
000001011 => 111110100 => [1,1,1,1,1,2,3] => ? = 3
000001100 => 111110011 => [1,1,1,1,1,3,1,1] => ? = 3
000001101 => 111110010 => [1,1,1,1,1,3,2] => ? = 3
000001110 => 111110001 => [1,1,1,1,1,4,1] => ? = 4
000001111 => 111110000 => [1,1,1,1,1,5] => ? = 5
000010000 => 111101111 => [1,1,1,1,2,1,1,1,1] => ? = 2
000010010 => 111101101 => [1,1,1,1,2,1,2,1] => ? = 2
000010011 => 111101100 => [1,1,1,1,2,1,3] => ? = 3
000010101 => 111101010 => [1,1,1,1,2,2,2] => ? = 2
000010110 => 111101001 => [1,1,1,1,2,3,1] => ? = 3
000010111 => 111101000 => [1,1,1,1,2,4] => ? = 4
000011000 => 111100111 => [1,1,1,1,3,1,1,1] => ? = 3
000011001 => 111100110 => [1,1,1,1,3,1,2] => ? = 3
000011010 => 111100101 => [1,1,1,1,3,2,1] => ? = 3
000011100 => 111100011 => [1,1,1,1,4,1,1] => ? = 4
000011101 => 111100010 => [1,1,1,1,4,2] => ? = 4
000011111 => 111100000 => [1,1,1,1,6] => ? = 6
000100000 => 111011111 => [1,1,1,2,1,1,1,1,1] => ? = 2
000100001 => 111011110 => [1,1,1,2,1,1,1,2] => ? = 2
000100010 => 111011101 => [1,1,1,2,1,1,2,1] => ? = 2
000100011 => 111011100 => [1,1,1,2,1,1,3] => ? = 3
000100100 => 111011011 => [1,1,1,2,1,2,1,1] => ? = 2
000100101 => 111011010 => [1,1,1,2,1,2,2] => ? = 2
000100110 => 111011001 => [1,1,1,2,1,3,1] => ? = 3
000100111 => 111011000 => [1,1,1,2,1,4] => ? = 4
000101000 => 111010111 => [1,1,1,2,2,1,1,1] => ? = 2
000101001 => 111010110 => [1,1,1,2,2,1,2] => ? = 2
000101010 => 111010101 => [1,1,1,2,2,2,1] => ? = 2
000101011 => 111010100 => [1,1,1,2,2,3] => ? = 3
000101100 => 111010011 => [1,1,1,2,3,1,1] => ? = 3
000101101 => 111010010 => [1,1,1,2,3,2] => ? = 3
000101110 => 111010001 => [1,1,1,2,4,1] => ? = 4
000101111 => 111010000 => [1,1,1,2,5] => ? = 5
000110000 => 111001111 => [1,1,1,3,1,1,1,1] => ? = 3
000110001 => 111001110 => [1,1,1,3,1,1,2] => ? = 3
000110010 => 111001101 => [1,1,1,3,1,2,1] => ? = 3
000110011 => 111001100 => [1,1,1,3,1,3] => ? = 3
000110100 => 111001011 => [1,1,1,3,2,1,1] => ? = 3
000110101 => 111001010 => [1,1,1,3,2,2] => ? = 3
000110110 => 111001001 => [1,1,1,3,3,1] => ? = 3
000110111 => 111001000 => [1,1,1,3,4] => ? = 4

Description

The largest part of an integer composition.

Matching statistic: St000982

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 83%

Values

0 => [2] => 10 => 01 => 1
1 => [1,1] => 11 => 11 => 2
00 => [3] => 100 => 101 => 1
01 => [2,1] => 101 => 001 => 2
10 => [1,2] => 110 => 011 => 2
11 => [1,1,1] => 111 => 111 => 3
000 => [4] => 1000 => 0101 => 1
001 => [3,1] => 1001 => 1101 => 2
010 => [2,2] => 1010 => 1001 => 2
011 => [2,1,1] => 1011 => 0001 => 3
100 => [1,3] => 1100 => 1011 => 2
101 => [1,2,1] => 1101 => 0011 => 2
110 => [1,1,2] => 1110 => 0111 => 3
111 => [1,1,1,1] => 1111 => 1111 => 4
0000 => [5] => 10000 => 10101 => 1
0001 => [4,1] => 10001 => 00101 => 2
0010 => [3,2] => 10010 => 01101 => 2
0011 => [3,1,1] => 10011 => 11101 => 3
0100 => [2,3] => 10100 => 01001 => 2
0101 => [2,2,1] => 10101 => 11001 => 2
0110 => [2,1,2] => 10110 => 10001 => 3
0111 => [2,1,1,1] => 10111 => 00001 => 4
1000 => [1,4] => 11000 => 01011 => 2
1001 => [1,3,1] => 11001 => 11011 => 2
1010 => [1,2,2] => 11010 => 10011 => 2
1011 => [1,2,1,1] => 11011 => 00011 => 3
1100 => [1,1,3] => 11100 => 10111 => 3
1101 => [1,1,2,1] => 11101 => 00111 => 3
1110 => [1,1,1,2] => 11110 => 01111 => 4
1111 => [1,1,1,1,1] => 11111 => 11111 => 5
00000 => [6] => 100000 => 010101 => 1
00001 => [5,1] => 100001 => 110101 => 2
00010 => [4,2] => 100010 => 100101 => 2
00011 => [4,1,1] => 100011 => 000101 => 3
00100 => [3,3] => 100100 => 101101 => 2
00101 => [3,2,1] => 100101 => 001101 => 2
00110 => [3,1,2] => 100110 => 011101 => 3
00111 => [3,1,1,1] => 100111 => 111101 => 4
01000 => [2,4] => 101000 => 101001 => 2
01001 => [2,3,1] => 101001 => 001001 => 2
01010 => [2,2,2] => 101010 => 011001 => 2
01011 => [2,2,1,1] => 101011 => 111001 => 3
01100 => [2,1,3] => 101100 => 010001 => 3
01101 => [2,1,2,1] => 101101 => 110001 => 3
01110 => [2,1,1,2] => 101110 => 100001 => 4
01111 => [2,1,1,1,1] => 101111 => 000001 => 5
10000 => [1,5] => 110000 => 101011 => 2
10001 => [1,4,1] => 110001 => 001011 => 2
10010 => [1,3,2] => 110010 => 011011 => 2
10011 => [1,3,1,1] => 110011 => 111011 => 3
000000001 => [9,1] => 1000000001 => 1101010101 => ? = 2
000000011 => [8,1,1] => 1000000011 => ? => ? = 3
000000100 => [7,3] => 1000000100 => 1011010101 => ? = 2
000000101 => [7,2,1] => 1000000101 => ? => ? = 2
000000110 => [7,1,2] => 1000000110 => 0111010101 => ? = 3
000000111 => [7,1,1,1] => 1000000111 => 1111010101 => ? = 4
000001000 => [6,4] => 1000001000 => ? => ? = 2
000001001 => [6,3,1] => 1000001001 => ? => ? = 2
000001011 => [6,2,1,1] => 1000001011 => ? => ? = 3
000001100 => [6,1,3] => 1000001100 => ? => ? = 3
000001110 => [6,1,1,2] => 1000001110 => 1000010101 => ? = 4
000001111 => [6,1,1,1,1] => 1000001111 => ? => ? = 5
000010000 => [5,5] => 1000010000 => ? => ? = 2
000010001 => [5,4,1] => 1000010001 => ? => ? = 2
000010010 => [5,3,2] => 1000010010 => ? => ? = 2
000010011 => [5,3,1,1] => 1000010011 => ? => ? = 3
000010100 => [5,2,3] => 1000010100 => ? => ? = 2
000010101 => [5,2,2,1] => 1000010101 => ? => ? = 2
000010110 => [5,2,1,2] => 1000010110 => 1000110101 => ? = 3
000010111 => [5,2,1,1,1] => 1000010111 => ? => ? = 4
000011000 => [5,1,4] => 1000011000 => ? => ? = 3
000011001 => [5,1,3,1] => 1000011001 => 1101110101 => ? = 3
000011010 => [5,1,2,2] => 1000011010 => ? => ? = 3
000011011 => [5,1,2,1,1] => 1000011011 => ? => ? = 3
000011100 => [5,1,1,3] => 1000011100 => ? => ? = 4
000011101 => [5,1,1,2,1] => 1000011101 => ? => ? = 4
000011111 => [5,1,1,1,1,1] => 1000011111 => 1111110101 => ? = 6
000100000 => [4,6] => 1000100000 => ? => ? = 2
000100001 => [4,5,1] => 1000100001 => ? => ? = 2
000100010 => [4,4,2] => 1000100010 => ? => ? = 2
000100011 => [4,4,1,1] => 1000100011 => ? => ? = 3
000100100 => [4,3,3] => 1000100100 => ? => ? = 2
000100110 => [4,3,1,2] => 1000100110 => ? => ? = 3
000100111 => [4,3,1,1,1] => 1000100111 => ? => ? = 4
000101000 => [4,2,4] => 1000101000 => ? => ? = 2
000101001 => [4,2,3,1] => 1000101001 => ? => ? = 2
000101010 => [4,2,2,2] => 1000101010 => ? => ? = 2
000101100 => [4,2,1,3] => 1000101100 => ? => ? = 3
000101101 => [4,2,1,2,1] => 1000101101 => ? => ? = 3
000101110 => [4,2,1,1,2] => 1000101110 => 0111100101 => ? = 4
000101111 => [4,2,1,1,1,1] => 1000101111 => ? => ? = 5
000110000 => [4,1,5] => 1000110000 => ? => ? = 3
000110010 => [4,1,3,2] => 1000110010 => ? => ? = 3
000110011 => [4,1,3,1,1] => 1000110011 => ? => ? = 3
000110100 => [4,1,2,3] => 1000110100 => ? => ? = 3
000110101 => [4,1,2,2,1] => 1000110101 => ? => ? = 3
000110111 => [4,1,2,1,1,1] => 1000110111 => 1111000101 => ? = 4
000111000 => [4,1,1,4] => 1000111000 => ? => ? = 4
000111001 => [4,1,1,3,1] => 1000111001 => ? => ? = 4
000111010 => [4,1,1,2,2] => 1000111010 => 0110000101 => ? = 4

Description

The length of the longest constant subword.

Matching statistic: St000013
(load all 2 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 83%

Values

0 => 1 => [1,1] => [1,0,1,0]
 => 1
1 => 0 => [2] => [1,1,0,0]
 => 2
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
11 => 00 => [3] => [1,1,1,0,0,0]
 => 3
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
0001010 => 1110101 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
0001011 => 1110100 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
0001100 => 1110011 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
0001110 => 1110001 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
0010010 => 1101101 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
0010011 => 1101100 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
0010100 => 1101011 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
0010110 => 1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
0011001 => 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 3
0011010 => 1100101 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 3
0011100 => 1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 4
0100010 => 1011101 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
0100011 => 1011100 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3
0100100 => 1011011 => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
0100101 => 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
0100110 => 1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
0101000 => 1010111 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
0101001 => 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
0101100 => 1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
0110001 => 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3
0110010 => 1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
0110100 => 1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
0111000 => 1000111 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
0111011 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
1000101 => 0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
1000110 => 0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
1001000 => 0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
1001001 => 0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
1001010 => 0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
1001100 => 0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
1010001 => 0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
1010010 => 0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
1010100 => 0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
1011011 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
1100010 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
1100100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
1101000 => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
1110011 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
1110110 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
1111001 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
00000010 => 11111101 => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
00000100 => 11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
00000110 => 11111001 => [1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
00001000 => 11110111 => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
00001001 => 11110110 => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
00001010 => 11110101 => [1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2

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
(load all 3 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 58%

Values

0 => 1 => [1,1] => [1,0,1,0]
 => 1
1 => 0 => [2] => [1,1,0,0]
 => 2
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
11 => 00 => [3] => [1,1,1,0,0,0]
 => 3
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
1100000 => 0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3
1100001 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
1100010 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
1100011 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3
1100100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
1100101 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
1100110 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
1101000 => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
1101001 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
1101010 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
1101100 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
1101110 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
1101111 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
1110000 => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
1110001 => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
1110010 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
1110011 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
1110100 => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
1110101 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
1110110 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
1110111 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
1111000 => 0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5
1111001 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
1111011 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
1111100 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6
1111101 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
1111110 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
1111111 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8
00000000 => 11111111 => [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
00000001 => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
00000010 => 11111101 => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
00000011 => 11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3
00000100 => 11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
00000101 => 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
00000110 => 11111001 => [1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
00000111 => 11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
00001000 => 11110111 => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
00001001 => 11110110 => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
00001010 => 11110101 => [1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
00001011 => 11110100 => [1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
00001100 => 11110011 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
00001101 => 11110010 => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
00001110 => 11110001 => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
00001111 => 11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
00010000 => 11101111 => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
00010001 => 11101110 => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2

Description

The length of the maximal rise of a Dyck path.

Matching statistic: St000442
(load all 2 compositions to match this statistic)

Mapping

Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 58%

Values

0 => 1 => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
1 => 0 => [2] => [1,1,0,0]
 => 1 = 2 - 1
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
11 => 00 => [3] => [1,1,1,0,0,0]
 => 2 = 3 - 1
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1100000 => 0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3 - 1
1100001 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 - 1
1100010 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 - 1
1100011 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 - 1
1100100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3 - 1
1100101 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
1100110 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
1101000 => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 - 1
1101001 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
1101010 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
1101100 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
1101110 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4 - 1
1101111 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 1
1110000 => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 - 1
1110001 => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 1
1110010 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 - 1
1110011 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
1110100 => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 - 1
1110101 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
1110110 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4 - 1
1110111 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
1111000 => 0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5 - 1
1111001 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5 - 1
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5 - 1
1111011 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5 - 1
1111100 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6 - 1
1111101 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6 - 1
1111110 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7 - 1
1111111 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 8 - 1
00000000 => 11111111 => [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 - 1
00000001 => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
00000010 => 11111101 => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
00000011 => 11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 - 1
00000100 => 11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
00000101 => 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
00000110 => 11111001 => [1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
00000111 => 11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
00001000 => 11110111 => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
00001001 => 11110110 => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
00001010 => 11110101 => [1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
00001011 => 11110100 => [1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
00001100 => 11110011 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1
00001101 => 11110010 => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
00001110 => 11110001 => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4 - 1
00001111 => 11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 1
00010000 => 11101111 => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
00010001 => 11101110 => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000684
(load all 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%

Values

0 => [2] => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 3
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 4
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3

Description

The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with

$n$ points for

$n \geq 2$ . Number those points from the left to the right by

$0,1,\ldots,n-1$ . The algebra is then uniquely determined by the dimension

$c_i$ of the projective indecomposable modules at point

$i$ . Such algebras are then uniquely determined by lists of the form

$[c_0,c_1,...,c_{n-1}]$ with the conditions:

$c_{n-1}=1$ and

$c_i -1 \leq c_{i+1}$ for all

$i$ . The number of such algebras is then the

$n-1$ -st Catalan number

$C_{n-1}$ . One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between

$n$ -LNakayama algebras with global dimension bounded by

$m$ and Dyck paths with height at most

$m$ . Examples: * For

$m=2$ , the number of Dyck paths with global dimension at most

$m$ starts for

$n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For

$m=3$ , the number of Dyck paths with global dimension at most

$m$ starts for

$n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.

Matching statistic: St001090

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%

Values

0 => [2] => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
1 => [1,1] => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
00 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 2 = 3 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 2 = 3 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1 = 2 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 2 = 3 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 3 = 4 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1 = 2 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 2 = 3 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 1 = 2 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 2 = 3 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 3 = 4 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 2 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 1 = 2 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 2 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 2 = 3 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 2 = 3 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 2 = 3 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 3 = 4 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 4 = 5 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0 = 1 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => 1 = 2 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,3,5,6,4] => 2 = 3 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,4,3,5,6] => 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5] => 1 = 2 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => 2 = 3 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,2,4,5,6,3] => 3 = 4 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,4,6,5] => 1 = 2 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,6] => 1 = 2 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,6,4] => 2 = 3 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => 2 = 3 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5] => 2 = 3 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,2,6] => 3 = 4 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,2] => 4 = 5 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => 1 = 2 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => 1 = 2 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,3,5,4,6] => 1 = 2 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,3,5,6,4] => 2 = 3 - 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 2 - 1
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 2 - 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => ? = 3 - 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2 - 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 2 - 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 3 - 1
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 4 - 1
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 2 - 1
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 2 - 1
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 2 - 1
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3 - 1
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => ? = 3 - 1
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3,7,6] => ? = 3 - 1
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => ? = 4 - 1
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,3] => ? = 5 - 1
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,3,1,4,5,7,6] => ? = 3 - 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,1,4,6,5,7] => ? = 3 - 1
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,3,1,4,6,7,5] => ? = 3 - 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,1,5,4,6,7] => ? = 3 - 1
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,7,6] => ? = 3 - 1
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => ? = 3 - 1
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,4] => ? = 4 - 1
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => ? = 4 - 1
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,1,6,7,5] => ? = 4 - 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,5,7,8,6] => ? = 3 - 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,2,3,4,6,5,7,8] => ? = 2 - 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,8,7] => ? = 2 - 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,2,3,4,6,7,5,8] => ? = 3 - 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,2,3,4,6,7,8,5] => ? = 4 - 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,2,3,5,4,6,8,7] => ? = 2 - 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,7,8,6] => ? = 3 - 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,2,3,5,6,4,7,8] => ? = 3 - 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,6,4,8,7] => ? = 3 - 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,6,7,4,8] => ? = 4 - 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,4,3,5,6,7,8] => ? = 2 - 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,4,3,5,6,8,7] => ? = 2 - 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5,7,6,8] => ? = 2 - 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,3,5,7,8,6] => ? = 3 - 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,8,7] => ? = 2 - 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,6,7,5,8] => ? = 3 - 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,6,7,8,5] => ? = 4 - 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,4,5,3,6,7,8] => ? = 3 - 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,4,5,3,6,8,7] => ? = 3 - 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6,8] => ? = 3 - 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,5,3,7,8,6] => ? = 3 - 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,4,5,6,3,7,8] => ? = 4 - 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,5,6,3,8,7] => ? = 4 - 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,4,5,6,7,3,8] => ? = 5 - 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,4,5,6,7,8,3] => ? = 6 - 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,3,2,4,5,6,8,7] => ? = 2 - 1

Description

The number of pop-stack-sorts needed to sort a permutation. The pop-stack sorting operator is defined as follows. Process the permutation

$\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order. A permutation is

$t$ -pop-stack sortable if it is sortable using

$t$ pop-stacks in series.

Matching statistic: St000686
(load all 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 58%

Values

0 => [2] => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 4
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 3
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 4
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3

Description

The finitistic dominant dimension of a Dyck path. To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.

The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000141The maximum drop size of a permutation. St000451The length of the longest pattern of the form k 1 2. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.