- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000290
(load all 3 compositions to match this statistic)

Mapping

St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0
=> 0
=> 0
=> 0
=> 1
=> 0
=> 0
=> 0
=> 2
=> 0
=> 1
=> 1
=> 2
=> 0
=> 0
=> 0
=> 3
=> 0
=> 2
=> 2
=> 3
=> 0
=> 1
=> 1
=> 4
=> 1
=> 2
=> 2
=> 3
=> 0
=> 0
=> 0
=> 4
=> 0
=> 3
=> 3
=> 4
=> 0
=> 2
=> 2
=> 6
=> 2
=> 3
=> 3
=> 4
=> 0
=> 1
=> 1
=> 5
=> 1

Description

The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length

$n$ with

$a$ zeros, the generating function for the major index is the

$q$ -binomial coefficient

$\binom{n}{a}_q$ .

Matching statistic: St000293
(load all 3 compositions to match this statistic)

Mapping

Mp00096: Binary words —Foata bijection⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 97%

Values

0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 01 => 0
10 => 10 => 1
11 => 11 => 0
000 => 000 => 0
001 => 001 => 0
010 => 100 => 2
011 => 011 => 0
100 => 010 => 1
101 => 101 => 1
110 => 110 => 2
111 => 111 => 0
0000 => 0000 => 0
0001 => 0001 => 0
0010 => 1000 => 3
0011 => 0011 => 0
0100 => 0100 => 2
0101 => 1001 => 2
0110 => 1010 => 3
0111 => 0111 => 0
1000 => 0010 => 1
1001 => 0101 => 1
1010 => 1100 => 4
1011 => 1011 => 1
1100 => 0110 => 2
1101 => 1101 => 2
1110 => 1110 => 3
1111 => 1111 => 0
00000 => 00000 => 0
00001 => 00001 => 0
00010 => 10000 => 4
00011 => 00011 => 0
00100 => 01000 => 3
00101 => 10001 => 3
00110 => 10010 => 4
00111 => 00111 => 0
01000 => 00100 => 2
01001 => 01001 => 2
01010 => 11000 => 6
01011 => 10011 => 2
01100 => 01010 => 3
01101 => 10101 => 3
01110 => 10110 => 4
01111 => 01111 => 0
10000 => 00010 => 1
10001 => 00101 => 1
10010 => 10100 => 5
10011 => 01011 => 1
10110111100000 => 00001110101110 => ? = 14
10111011010000 => 00011110110100 => ? = 24
10111011100000 => 00001110110110 => ? = 15
10111100110000 => 00010110111010 => ? = 17
10111101001000 => 00101110111000 => ? = 26
10111101010000 => 00011110111000 => ? = 25
10111101100000 => 00001110111010 => ? = 16
10111110000100 => 01000110111100 => ? = 20
10111110001000 => 00100110111100 => ? = 19
10111110010000 => 00010110111100 => ? = 18
10111110100000 => 00001110111100 => ? = 17
10111111000000 => 00000110111110 => ? = 9
11001111100000 => 00001011011110 => ? = 11
11011110000010 => 10000111011100 => ? = 22
11101101000010 => 10001111101000 => ? = 30
11101110000010 => 10000111101100 => ? = 23
11110011000010 => 10001011110100 => ? = 25
11110100100010 => 10010111110000 => ? = 32
11110101000010 => 10001111110000 => ? = 31
11110110000010 => 10000111110100 => ? = 24
11111000001100 => 01000011111010 => ? = 17
11111000010010 => 10100011111000 => ? = 28
11111000100010 => 10010011111000 => ? = 27
11111001000010 => 10001011111000 => ? = 26
11111010000010 => 10000111111000 => ? = 25
11111100000010 => 10000011111100 => ? = 19
1011110111000000 => 0000011101110110 => ? = 17
1011111010100000 => 0000111101111000 => ? = 28
1011111011000000 => 0000011101111010 => ? = 18
1011111100010000 => 0001001101111100 => ? = 21
1011111100100000 => 0000101101111100 => ? = 20
1011111101000000 => 0000011101111100 => ? = 19
1011111110000000 => 0000001101111110 => ? = 10
1111011100000010 => 1000001111101100 => ? = 27
1111101010000010 => 1000011111110000 => ? = 36
1111101100000010 => 1000001111110100 => ? = 28
1111110001000010 => 1000100111111000 => ? = 31
1111110010000010 => 1000010111111000 => ? = 30
1111110100000010 => 1000001111111000 => ? = 29
1111111000000010 => 1000000111111100 => ? = 22
1110111110 => 1111011110 => ? = 12
1110011110 => 1011101110 => ? = 12
1110010001 => 0010111001 => ? = 9
1110001001 => 0100111001 => ? = 10
1110000101 => 1000111001 => ? = 11
1101010001 => 0011110001 => ? = 12
1101001110 => 1011100110 => ? = 15
1100010010 => 1010011000 => ? = 17
1011111101 => 1101111101 => ? = 9
1011010001 => 0011101001 => ? = 11

Description

The number of inversions of a binary word.

Matching statistic: St000597

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000597: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 26%

Values

0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 0
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2,3,4},{5},{6}}
 => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6}}
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 5
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 5
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 6
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 4
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 10
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 4
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 5
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4},{5},{6,7},{8}}
 => ? = 5
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 6
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3},{4,5,6,7,8}}
 => ? = 3
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3},{4,5,6,7},{8}}
 => ? = 3
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 9
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 3
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5},{6,7,8}}
 => ? = 8
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 8
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4,5},{6},{7,8}}
 => ? = 9
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4,5},{6},{7},{8}}
 => ? = 3
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 4
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 4
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => {{1,2,3},{4},{5,6},{7,8}}
 => ? = 10
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2,3},{4},{5,6},{7},{8}}
 => ? = 4
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 5
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2,3},{4},{5},{6,7},{8}}
 => ? = 5
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5},{6},{7,8}}
 => ? = 6
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6},{7},{8}}
 => ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6},{7,8}}
 => ? = 8
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2},{3,4,5,6},{7},{8}}
 => ? = 2
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5},{6,7,8}}
 => ? = 7
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4,5},{6,7},{8}}
 => ? = 7
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2},{3,4,5},{6},{7,8}}
 => ? = 8
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2},{3,4,5},{6},{7},{8}}
 => ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 6
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4},{5,6,7},{8}}
 => ? = 6
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 12
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 6
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3,4},{5},{6,7,8}}
 => ? = 7
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5},{6,7},{8}}
 => ? = 7
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3,4},{5},{6},{7,8}}
 => ? = 8
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 2
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 3
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2},{3},{4,5,6,7},{8}}
 => ? = 3
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2},{3},{4,5,6},{7,8}}
 => ? = 9
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2},{3},{4,5,6},{7},{8}}
 => ? = 3
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5},{6,7,8}}
 => ? = 8
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6,7},{8}}
 => ? = 8

Description

The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block.

Matching statistic: St000605

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 26%

Values

0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 0
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2,3,4},{5},{6}}
 => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6}}
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 5
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 5
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 6
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 4
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 10
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 4
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 5
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4},{5},{6,7},{8}}
 => ? = 5
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 6
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3},{4,5,6,7,8}}
 => ? = 3
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3},{4,5,6,7},{8}}
 => ? = 3
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 9
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 3
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5},{6,7,8}}
 => ? = 8
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 8
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4,5},{6},{7,8}}
 => ? = 9
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4,5},{6},{7},{8}}
 => ? = 3
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 4
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 4
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => {{1,2,3},{4},{5,6},{7,8}}
 => ? = 10
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2,3},{4},{5,6},{7},{8}}
 => ? = 4
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 5
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2,3},{4},{5},{6,7},{8}}
 => ? = 5
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5},{6},{7,8}}
 => ? = 6
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6},{7},{8}}
 => ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6},{7,8}}
 => ? = 8
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2},{3,4,5,6},{7},{8}}
 => ? = 2
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5},{6,7,8}}
 => ? = 7
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4,5},{6,7},{8}}
 => ? = 7
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2},{3,4,5},{6},{7,8}}
 => ? = 8
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2},{3,4,5},{6},{7},{8}}
 => ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 6
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4},{5,6,7},{8}}
 => ? = 6
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6},{7,8}}
 => ? = 12
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 6
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3,4},{5},{6,7,8}}
 => ? = 7
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5},{6,7},{8}}
 => ? = 7
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3,4},{5},{6},{7,8}}
 => ? = 8
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 2
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 3
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2},{3},{4,5,6,7},{8}}
 => ? = 3
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2},{3},{4,5,6},{7,8}}
 => ? = 9
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2},{3},{4,5,6},{7},{8}}
 => ? = 3
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5},{6,7,8}}
 => ? = 8
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6,7},{8}}
 => ? = 8

Description

The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block.

Matching statistic: St000462

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 26%

Values

0 => [2] => [1,1,0,0]
 => [2,1] => 0
1 => [1,1] => [1,0,1,0]
 => [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,1,2,3,6,5] => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,5,6] => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,1,2,6,4,5] => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,1,2,5,4,6] => 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2,4,6,5] => 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,2,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,3,4,5] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,3,4,6] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,4,5] => 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,2,3,4,5] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,2,3,4,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,2,3,6,5] => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,2,3,5,6] => 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 0
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6] => ? = 5
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,1,2,3,4,6,7] => ? = 0
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 4
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 4
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,1,2,3,5,7,6] => ? = 5
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,1,2,3,5,6,7] => ? = 0
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 3
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 3
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 8
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 3
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 4
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 4
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,1,2,4,5,7,6] => ? = 5
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,1,2,4,5,6,7] => ? = 0
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,3,4,5,6] => ? = 2
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,3,4,5,7] => ? = 2
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,3,4,7,6] => ? = 7
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,3,4,6,7] => ? = 2
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,5,6] => ? = 6
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? = 6
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => ? = 7
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => ? = 2
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,4,5,6] => ? = 3
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,4,5,7] => ? = 3
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => ? = 8
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => ? = 3
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,5,6] => ? = 4
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => ? = 4
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? = 5
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7] => ? = 0
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,2,3,4,5,6] => ? = 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,2,3,4,5,7] => ? = 1
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,2,3,4,7,6] => ? = 6
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,2,3,4,6,7] => ? = 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [8,1,2,3,4,5,6,7] => ? = 0
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,8] => ? = 0
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [6,1,2,3,4,5,8,7] => ? = 6
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [6,1,2,3,4,5,7,8] => ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [5,1,2,3,4,8,6,7] => ? = 5
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,4,7,6,8] => ? = 5
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [5,1,2,3,4,6,8,7] => ? = 6
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [5,1,2,3,4,6,7,8] => ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [4,1,2,3,8,5,6,7] => ? = 4
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,3,7,5,6,8] => ? = 4
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [4,1,2,3,6,5,8,7] => ? = 10
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [4,1,2,3,6,5,7,8] => ? = 4
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [4,1,2,3,5,8,6,7] => ? = 5
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [4,1,2,3,5,7,6,8] => ? = 5

Description

The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].

Matching statistic: St001811

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 13%

Values

0 => [2] => [1,1,0,0]
 => [2,1] => 0
1 => [1,1] => [1,0,1,0]
 => [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => ? = 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => ? = 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,1,2,3,6,5] => ? = 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,5,6] => ? = 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,1,2,6,4,5] => ? = 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,1,2,5,4,6] => ? = 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2,4,6,5] => ? = 4
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,2,4,5,6] => ? = 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,3,4,5] => ? = 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,3,4,6] => ? = 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => ? = 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => ? = 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,4,5] => ? = 3
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => ? = 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => ? = 4
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => ? = 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,2,3,4,5] => ? = 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,2,3,4,6] => ? = 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,2,3,6,5] => ? = 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,2,3,5,6] => ? = 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,4,5] => ? = 4
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => ? = 4
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => ? = 5
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => ? = 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => ? = 2
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,3,4,6] => ? = 2
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ? = 6
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ? = 2
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => ? = 3
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ? = 3
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ? = 4
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? = 0
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => ? = 0
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,1,2,3,4,7,6] => ? = 5
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,1,2,3,4,6,7] => ? = 0
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 4
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 4
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,1,2,3,5,7,6] => ? = 5
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,1,2,3,5,6,7] => ? = 0
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 3
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 3
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 8
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 3
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 4
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 4
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,1,2,4,5,7,6] => ? = 5
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,1,2,4,5,6,7] => ? = 0
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,3,4,5,6] => ? = 2
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,3,4,5,7] => ? = 2

Description

The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation

$\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety''

$X_\sigma$ . Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for

$\sigma$ . It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].