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
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 2
011 => 0
100 => 1
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 3
0011 => 0
0100 => 2
0101 => 2
0110 => 3
0111 => 0
1000 => 1
1001 => 1
1010 => 4
1011 => 1
1100 => 2
1101 => 2
1110 => 3
1111 => 0
00000 => 0
00001 => 0
00010 => 4
00011 => 0
00100 => 3
00101 => 3
00110 => 4
00111 => 0
01000 => 2
01001 => 2
01010 => 6
01011 => 2
01100 => 3
01101 => 3
01110 => 4
01111 => 0
10000 => 1
10001 => 1
10010 => 5
10011 => 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The number of inversions of a binary word.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: 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: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
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
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,4,2,3,7,5,6] => ? = 5
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,4,2,3,6,5,7] => ? = 5
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,4,2,3,5,7,6] => ? = 6
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,4,2,3,5,6,7] => ? = 1
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,7,4,5,6] => ? = 4
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,7] => ? = 4
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => ? = 9
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => ? = 4
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,5,6] => ? = 5
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => ? = 5
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => ? = 6
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => ? = 1
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,3,4,5,6] => ? = 2
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,3,4,5,7] => ? = 2
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,3,4,7,6] => ? = 7
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => ? = 2
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,5,6] => ? = 6
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => ? = 6
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]].