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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000142
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> 1
1 => [1,1] => [1,1]
=> 0
00 => [3] => [3]
=> 0
01 => [2,1] => [2,1]
=> 1
10 => [1,2] => [2,1]
=> 1
11 => [1,1,1] => [1,1,1]
=> 0
000 => [4] => [4]
=> 1
001 => [3,1] => [3,1]
=> 0
010 => [2,2] => [2,2]
=> 2
011 => [2,1,1] => [2,1,1]
=> 1
100 => [1,3] => [3,1]
=> 0
101 => [1,2,1] => [2,1,1]
=> 1
110 => [1,1,2] => [2,1,1]
=> 1
111 => [1,1,1,1] => [1,1,1,1]
=> 0
0000 => [5] => [5]
=> 0
0001 => [4,1] => [4,1]
=> 1
0010 => [3,2] => [3,2]
=> 1
0011 => [3,1,1] => [3,1,1]
=> 0
0100 => [2,3] => [3,2]
=> 1
0101 => [2,2,1] => [2,2,1]
=> 2
0110 => [2,1,2] => [2,2,1]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> 1
1000 => [1,4] => [4,1]
=> 1
1001 => [1,3,1] => [3,1,1]
=> 0
1010 => [1,2,2] => [2,2,1]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> 1
1100 => [1,1,3] => [3,1,1]
=> 0
1101 => [1,1,2,1] => [2,1,1,1]
=> 1
1110 => [1,1,1,2] => [2,1,1,1]
=> 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
00000 => [6] => [6]
=> 1
00001 => [5,1] => [5,1]
=> 0
00010 => [4,2] => [4,2]
=> 2
00011 => [4,1,1] => [4,1,1]
=> 1
00100 => [3,3] => [3,3]
=> 0
00101 => [3,2,1] => [3,2,1]
=> 1
00110 => [3,1,2] => [3,2,1]
=> 1
00111 => [3,1,1,1] => [3,1,1,1]
=> 0
01000 => [2,4] => [4,2]
=> 2
01001 => [2,3,1] => [3,2,1]
=> 1
01010 => [2,2,2] => [2,2,2]
=> 3
01011 => [2,2,1,1] => [2,2,1,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> 1
01101 => [2,1,2,1] => [2,2,1,1]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1
10000 => [1,5] => [5,1]
=> 0
10001 => [1,4,1] => [4,1,1]
=> 1
10010 => [1,3,2] => [3,2,1]
=> 1
10011 => [1,3,1,1] => [3,1,1,1]
=> 0
Description
The number of even parts of a partition.
Matching statistic: St000149
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [2]
=> 1
1 => [1,1] => [1,1]
=> [1,1]
=> 0
00 => [3] => [3]
=> [2,1]
=> 0
01 => [2,1] => [2,1]
=> [3]
=> 1
10 => [1,2] => [2,1]
=> [3]
=> 1
11 => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 0
000 => [4] => [4]
=> [2,2]
=> 1
001 => [3,1] => [3,1]
=> [2,1,1]
=> 0
010 => [2,2] => [2,2]
=> [4]
=> 2
011 => [2,1,1] => [2,1,1]
=> [3,1]
=> 1
100 => [1,3] => [3,1]
=> [2,1,1]
=> 0
101 => [1,2,1] => [2,1,1]
=> [3,1]
=> 1
110 => [1,1,2] => [2,1,1]
=> [3,1]
=> 1
111 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 0
0000 => [5] => [5]
=> [2,2,1]
=> 0
0001 => [4,1] => [4,1]
=> [3,2]
=> 1
0010 => [3,2] => [3,2]
=> [4,1]
=> 1
0011 => [3,1,1] => [3,1,1]
=> [2,1,1,1]
=> 0
0100 => [2,3] => [3,2]
=> [4,1]
=> 1
0101 => [2,2,1] => [2,2,1]
=> [5]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [5]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [3,1,1]
=> 1
1000 => [1,4] => [4,1]
=> [3,2]
=> 1
1001 => [1,3,1] => [3,1,1]
=> [2,1,1,1]
=> 0
1010 => [1,2,2] => [2,2,1]
=> [5]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> 1
1100 => [1,1,3] => [3,1,1]
=> [2,1,1,1]
=> 0
1101 => [1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> 1
1110 => [1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
00000 => [6] => [6]
=> [2,2,2]
=> 1
00001 => [5,1] => [5,1]
=> [2,2,1,1]
=> 0
00010 => [4,2] => [4,2]
=> [4,2]
=> 2
00011 => [4,1,1] => [4,1,1]
=> [4,1,1]
=> 1
00100 => [3,3] => [3,3]
=> [3,2,1]
=> 0
00101 => [3,2,1] => [3,2,1]
=> [3,3]
=> 1
00110 => [3,1,2] => [3,2,1]
=> [3,3]
=> 1
00111 => [3,1,1,1] => [3,1,1,1]
=> [2,1,1,1,1]
=> 0
01000 => [2,4] => [4,2]
=> [4,2]
=> 2
01001 => [2,3,1] => [3,2,1]
=> [3,3]
=> 1
01010 => [2,2,2] => [2,2,2]
=> [6]
=> 3
01011 => [2,2,1,1] => [2,2,1,1]
=> [5,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [3,3]
=> 1
01101 => [2,1,2,1] => [2,2,1,1]
=> [5,1]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [5,1]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [3,1,1,1]
=> 1
10000 => [1,5] => [5,1]
=> [2,2,1,1]
=> 0
10001 => [1,4,1] => [4,1,1]
=> [4,1,1]
=> 1
10010 => [1,3,2] => [3,2,1]
=> [3,3]
=> 1
10011 => [1,3,1,1] => [3,1,1,1]
=> [2,1,1,1,1]
=> 0
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000150
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [1,1]
=> 1
1 => [1,1] => [1,1]
=> [2]
=> 0
00 => [3] => [3]
=> [3]
=> 0
01 => [2,1] => [2,1]
=> [1,1,1]
=> 1
10 => [1,2] => [2,1]
=> [1,1,1]
=> 1
11 => [1,1,1] => [1,1,1]
=> [2,1]
=> 0
000 => [4] => [4]
=> [2,2]
=> 1
001 => [3,1] => [3,1]
=> [3,1]
=> 0
010 => [2,2] => [2,2]
=> [1,1,1,1]
=> 2
011 => [2,1,1] => [2,1,1]
=> [2,1,1]
=> 1
100 => [1,3] => [3,1]
=> [3,1]
=> 0
101 => [1,2,1] => [2,1,1]
=> [2,1,1]
=> 1
110 => [1,1,2] => [2,1,1]
=> [2,1,1]
=> 1
111 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 0
0000 => [5] => [5]
=> [5]
=> 0
0001 => [4,1] => [4,1]
=> [2,2,1]
=> 1
0010 => [3,2] => [3,2]
=> [3,1,1]
=> 1
0011 => [3,1,1] => [3,1,1]
=> [3,2]
=> 0
0100 => [2,3] => [3,2]
=> [3,1,1]
=> 1
0101 => [2,2,1] => [2,2,1]
=> [1,1,1,1,1]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [1,1,1,1,1]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1000 => [1,4] => [4,1]
=> [2,2,1]
=> 1
1001 => [1,3,1] => [3,1,1]
=> [3,2]
=> 0
1010 => [1,2,2] => [2,2,1]
=> [1,1,1,1,1]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1100 => [1,1,3] => [3,1,1]
=> [3,2]
=> 0
1101 => [1,1,2,1] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1110 => [1,1,1,2] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [4,1]
=> 0
00000 => [6] => [6]
=> [3,3]
=> 1
00001 => [5,1] => [5,1]
=> [5,1]
=> 0
00010 => [4,2] => [4,2]
=> [2,2,1,1]
=> 2
00011 => [4,1,1] => [4,1,1]
=> [2,2,2]
=> 1
00100 => [3,3] => [3,3]
=> [6]
=> 0
00101 => [3,2,1] => [3,2,1]
=> [3,1,1,1]
=> 1
00110 => [3,1,2] => [3,2,1]
=> [3,1,1,1]
=> 1
00111 => [3,1,1,1] => [3,1,1,1]
=> [3,2,1]
=> 0
01000 => [2,4] => [4,2]
=> [2,2,1,1]
=> 2
01001 => [2,3,1] => [3,2,1]
=> [3,1,1,1]
=> 1
01010 => [2,2,2] => [2,2,2]
=> [1,1,1,1,1,1]
=> 3
01011 => [2,2,1,1] => [2,2,1,1]
=> [2,1,1,1,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [3,1,1,1]
=> 1
01101 => [2,1,2,1] => [2,2,1,1]
=> [2,1,1,1,1]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [2,1,1,1,1]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [4,1,1]
=> 1
10000 => [1,5] => [5,1]
=> [5,1]
=> 0
10001 => [1,4,1] => [4,1,1]
=> [2,2,2]
=> 1
10010 => [1,3,2] => [3,2,1]
=> [3,1,1,1]
=> 1
10011 => [1,3,1,1] => [3,1,1,1]
=> [3,2,1]
=> 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000237
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: 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] => 1
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
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]
=> [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 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] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
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] => 1
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]
=> [5,4,3,2,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
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] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
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]
=> [1,2,5,4,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
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]
=> [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 3
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]
=> [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0
Description
The number of small exceedances.
This is the number of indices i such that πi=i+1.
Matching statistic: St001465
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001465: 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] => 1
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
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]
=> [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 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] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
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] => 1
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]
=> [5,4,3,2,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
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] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
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]
=> [1,2,5,4,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
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]
=> [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 3
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]
=> [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000389
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00105: Binary words —complement⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St000389: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1
1 => 0 => 0
00 => 11 => 0
01 => 10 => 1
10 => 01 => 1
11 => 00 => 0
000 => 111 => 1
001 => 110 => 0
010 => 101 => 2
011 => 100 => 1
100 => 011 => 0
101 => 010 => 1
110 => 001 => 1
111 => 000 => 0
0000 => 1111 => 0
0001 => 1110 => 1
0010 => 1101 => 1
0011 => 1100 => 0
0100 => 1011 => 1
0101 => 1010 => 2
0110 => 1001 => 2
0111 => 1000 => 1
1000 => 0111 => 1
1001 => 0110 => 0
1010 => 0101 => 2
1011 => 0100 => 1
1100 => 0011 => 0
1101 => 0010 => 1
1110 => 0001 => 1
1111 => 0000 => 0
00000 => 11111 => 1
00001 => 11110 => 0
00010 => 11101 => 2
00011 => 11100 => 1
00100 => 11011 => 0
00101 => 11010 => 1
00110 => 11001 => 1
00111 => 11000 => 0
01000 => 10111 => 2
01001 => 10110 => 1
01010 => 10101 => 3
01011 => 10100 => 2
01100 => 10011 => 1
01101 => 10010 => 2
01110 => 10001 => 2
01111 => 10000 => 1
10000 => 01111 => 0
10001 => 01110 => 1
10010 => 01101 => 1
10011 => 01100 => 0
=> => ? = 0
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St001230
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 1
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 1
101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 0
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
1010111 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
1011101 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
1011111 => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
1101011 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
1110101 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
00000111 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
00001111 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
00011111 => [4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
00111111 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
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