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Your data matches 7 different statistics following compositions of up to 3 maps.
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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: 93% ●values known / values provided: 93%●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: 93% ●values known / values provided: 93%●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
101011001100 => [1,2,2,1,3,1,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101011100100 => [1,2,2,1,1,3,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101100101100 => [1,2,1,3,2,1,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101100110010 => [1,2,1,3,1,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101100110100 => [1,2,1,3,1,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101101001100 => [1,2,1,2,3,1,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101101100100 => [1,2,1,2,1,3,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101110010010 => [1,2,1,1,3,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101110010100 => [1,2,1,1,3,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
101110100100 => [1,2,1,1,2,3,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110010101100 => [1,1,3,2,2,1,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110010110010 => [1,1,3,2,1,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110010110100 => [1,1,3,2,1,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110011001010 => [1,1,3,1,3,2,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110011010010 => [1,1,3,1,2,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110011010100 => [1,1,3,1,2,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110100101100 => [1,1,2,3,2,1,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110100110010 => [1,1,2,3,1,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110100110100 => [1,1,2,3,1,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110101001100 => [1,1,2,2,3,1,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110101100100 => [1,1,2,2,1,3,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110110010010 => [1,1,2,1,3,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110110010100 => [1,1,2,1,3,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
110110100100 => [1,1,2,1,2,3,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
111001001010 => [1,1,1,3,3,2,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
111001010010 => [1,1,1,3,2,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
111001010100 => [1,1,1,3,2,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
111010010010 => [1,1,1,2,3,3,2] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
111010010100 => [1,1,1,2,3,2,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
111010100100 => [1,1,1,2,2,3,3] => [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
1111101101 => [1,1,1,1,1,2,1,2,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1111101011 => [1,1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1110111101 => [1,1,1,2,1,1,1,2,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1110110111 => [1,1,1,2,1,2,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1101111110 => [1,1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1101111011 => [1,1,2,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1101011111 => [1,1,2,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1011111101 => [1,2,1,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1011110111 => [1,2,1,1,1,2,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1011011111 => [1,2,1,2,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
1001000000 => [1,3,7] => [7,3,1]
=> [7,3,1]
=> ? = 0
1000000100 => [1,7,3] => [7,3,1]
=> [7,3,1]
=> ? = 0
10001111110 => [1,4,1,1,1,1,1,2] => [4,2,1,1,1,1,1,1]
=> [4,2,2,2,1,1]
=> ? = 2
10011111110 => [1,3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 1
0111111101 => [2,1,1,1,1,1,1,2,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
0111111011 => [2,1,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
01111111001 => [2,1,1,1,1,1,1,3,1] => [3,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 1
01111110001 => [2,1,1,1,1,1,4,1] => [4,2,1,1,1,1,1,1]
=> [4,2,2,2,1,1]
=> ? = 2
0010000001 => [3,7,1] => [7,3,1]
=> [7,3,1]
=> ? = 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: 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: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
St000389: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 79%●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
1101001011 => 0010110100 => ? = 2
1100101011 => 0011010100 => ? = 2
1100000001 => 0011111110 => ? = 1
1100000000 => 0011111111 => ? = 0
1011110111 => 0100001000 => ? = 2
1011110001 => 0100001110 => ? = 2
1011100101 => 0100011010 => ? = 2
1011011111 => 0100100000 => ? = 2
1011011001 => 0100100110 => ? = 2
1011001101 => 0100110010 => ? = 2
1011000111 => 0100111000 => ? = 2
1011000001 => 0100111110 => ? = 2
1010110101 => 0101001010 => ? = 4
1010101011 => 0101010100 => ? = 4
1010011101 => 0101100010 => ? = 2
1010010111 => 0101101000 => ? = 2
1010010001 => 0101101110 => ? = 2
1010000101 => ? => ? = 2
1010000001 => 0101111110 => ? = 1
1001111111 => 0110000000 => ? = 0
1001111001 => ? => ? = 0
1001101101 => 0110010010 => ? = 2
1001100111 => 0110011000 => ? = 0
1001100001 => 0110011110 => ? = 0
1000111101 => 0111000010 => ? = 2
1000110111 => 0111001000 => ? = 2
1000110001 => 0111001110 => ? = 2
1000100101 => 0111011010 => ? = 2
1000100000 => 0111011111 => ? = 2
1000011111 => 0111100000 => ? = 0
1000011001 => ? => ? = 0
1000010000 => 0111101111 => ? = 0
1000001101 => 0111110010 => ? = 2
1000001001 => 0111110110 => ? = 1
1000000111 => 0111111000 => ? = 0
1000000101 => 0111111010 => ? = 1
1000000011 => 0111111100 => ? = 1
1000000001 => 0111111110 => ? = 0
10000000100 => 01111111011 => ? = 1
11011111110 => 00100000001 => ? = 2
10000010000 => 01111101111 => ? = 1
10000100010 => 01111011101 => ? = 2
10001000010 => ? => ? = 2
10000100100 => 01111011011 => ? = 0
10001001010 => 01110110101 => ? = 3
10101101110 => ? => ? = 4
11011011110 => ? => ? = 3
10001000100 => 01110111011 => ? = 2
10000101000 => 01111010111 => ? = 2
10010001010 => ? => ? = 3
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000142
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●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
1010111000 => [1,2,2,1,1,4] => [4,2,2,1,1,1]
=> ? = 3
1011001100 => [1,2,1,3,1,3] => [3,3,2,1,1,1]
=> ? = 1
1011011000 => [1,2,1,2,1,4] => [4,2,2,1,1,1]
=> ? = 3
1011100010 => [1,2,1,1,4,2] => [4,2,2,1,1,1]
=> ? = 3
1011100100 => [1,2,1,1,3,3] => [3,3,2,1,1,1]
=> ? = 1
1011101000 => [1,2,1,1,2,4] => [4,2,2,1,1,1]
=> ? = 3
1100101100 => [1,1,3,2,1,3] => [3,3,2,1,1,1]
=> ? = 1
1100110010 => [1,1,3,1,3,2] => [3,3,2,1,1,1]
=> ? = 1
1100110100 => [1,1,3,1,2,3] => [3,3,2,1,1,1]
=> ? = 1
1100111000 => [1,1,3,1,1,4] => [4,3,1,1,1,1]
=> ? = 1
1101001100 => [1,1,2,3,1,3] => [3,3,2,1,1,1]
=> ? = 1
1101011000 => [1,1,2,2,1,4] => [4,2,2,1,1,1]
=> ? = 3
1101100010 => [1,1,2,1,4,2] => [4,2,2,1,1,1]
=> ? = 3
1101100100 => [1,1,2,1,3,3] => [3,3,2,1,1,1]
=> ? = 1
1101101000 => [1,1,2,1,2,4] => [4,2,2,1,1,1]
=> ? = 3
1110001010 => [1,1,1,4,2,2] => [4,2,2,1,1,1]
=> ? = 3
1110001100 => [1,1,1,4,1,3] => [4,3,1,1,1,1]
=> ? = 1
1110010010 => [1,1,1,3,3,2] => [3,3,2,1,1,1]
=> ? = 1
1110010100 => [1,1,1,3,2,3] => [3,3,2,1,1,1]
=> ? = 1
1110011000 => [1,1,1,3,1,4] => [4,3,1,1,1,1]
=> ? = 1
1110100010 => [1,1,1,2,4,2] => [4,2,2,1,1,1]
=> ? = 3
1110100100 => [1,1,1,2,3,3] => [3,3,2,1,1,1]
=> ? = 1
1110101000 => [1,1,1,2,2,4] => [4,2,2,1,1,1]
=> ? = 3
1111000100 => [1,1,1,1,4,3] => [4,3,1,1,1,1]
=> ? = 1
1111001000 => [1,1,1,1,3,4] => [4,3,1,1,1,1]
=> ? = 1
101011001100 => [1,2,2,1,3,1,3] => [3,3,2,2,1,1,1]
=> ? = 2
101011100100 => [1,2,2,1,1,3,3] => [3,3,2,2,1,1,1]
=> ? = 2
101011110000 => [1,2,2,1,1,1,5] => [5,2,2,1,1,1,1]
=> ? = 2
101100101100 => [1,2,1,3,2,1,3] => [3,3,2,2,1,1,1]
=> ? = 2
101100110010 => [1,2,1,3,1,3,2] => [3,3,2,2,1,1,1]
=> ? = 2
101100110100 => [1,2,1,3,1,2,3] => [3,3,2,2,1,1,1]
=> ? = 2
101101001100 => [1,2,1,2,3,1,3] => [3,3,2,2,1,1,1]
=> ? = 2
101101100100 => [1,2,1,2,1,3,3] => [3,3,2,2,1,1,1]
=> ? = 2
101101110000 => [1,2,1,2,1,1,5] => [5,2,2,1,1,1,1]
=> ? = 2
101110010010 => [1,2,1,1,3,3,2] => [3,3,2,2,1,1,1]
=> ? = 2
101110010100 => [1,2,1,1,3,2,3] => [3,3,2,2,1,1,1]
=> ? = 2
101110100100 => [1,2,1,1,2,3,3] => [3,3,2,2,1,1,1]
=> ? = 2
101110110000 => [1,2,1,1,2,1,5] => [5,2,2,1,1,1,1]
=> ? = 2
101111000010 => [1,2,1,1,1,5,2] => [5,2,2,1,1,1,1]
=> ? = 2
101111010000 => [1,2,1,1,1,2,5] => [5,2,2,1,1,1,1]
=> ? = 2
110010101100 => [1,1,3,2,2,1,3] => [3,3,2,2,1,1,1]
=> ? = 2
110010110010 => [1,1,3,2,1,3,2] => [3,3,2,2,1,1,1]
=> ? = 2
110010110100 => [1,1,3,2,1,2,3] => [3,3,2,2,1,1,1]
=> ? = 2
110011001010 => [1,1,3,1,3,2,2] => [3,3,2,2,1,1,1]
=> ? = 2
110011001100 => [1,1,3,1,3,1,3] => [3,3,3,1,1,1,1]
=> ? = 0
110011010010 => [1,1,3,1,2,3,2] => [3,3,2,2,1,1,1]
=> ? = 2
110011010100 => [1,1,3,1,2,2,3] => [3,3,2,2,1,1,1]
=> ? = 2
110011100100 => [1,1,3,1,1,3,3] => [3,3,3,1,1,1,1]
=> ? = 0
110011110000 => [1,1,3,1,1,1,5] => [5,3,1,1,1,1,1]
=> ? = 0
110100101100 => [1,1,2,3,2,1,3] => [3,3,2,2,1,1,1]
=> ? = 2
Description
The number of even parts of a partition.
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: 10% ●values known / values provided: 10%●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: 10% ●values known / values provided: 10%●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
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 2
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 0
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 0
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 2
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => ? = 2
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
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] => ? = 3
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] => ? = 3
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]
=> [2,1,3,7,6,5,4] => ? = 2
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => ? = 1
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] => ? = 3
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] => ? = 2
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => ? = 1
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] => ? = 2
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] => ? = 2
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 0
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 2
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,8] => ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,4,3,2,1,6,8,7] => ? = 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,2,1,7,6,5,8] => ? = 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,3,2,1,6,5,7,8] => ? = 2
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,3,2,1,5,8,7,6] => ? = 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [4,3,2,1,5,7,6,8] => ? = 2
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,3,2,1,5,6,8,7] => ? = 2
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,2,1,7,6,5,4,8] => ? = 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => ? = 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8] => ? = 0
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,1,5,4,8,7,6] => ? = 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,7,6,8] => ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,5,4,6,8,7] => ? = 2
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8] => ? = 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,2,1,4,8,7,6,5] => ? = 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,2,1,4,7,6,5,8] => ? = 0
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,2,1,4,6,5,8,7] => ? = 2
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,2,1,4,6,5,7,8] => ? = 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,5,7,6,8] => ? = 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,6,8,7] => ? = 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,6,5,4,3,8] => ? = 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ? = 2
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3,8,7,6] => ? = 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
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: 8% ●values known / values provided: 8%●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: 8% ●values known / values provided: 8%●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
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 2
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 2
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 0
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 0
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 2
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => ? = 2
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 2
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] => ? = 3
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] => ? = 3
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]
=> [2,1,3,7,6,5,4] => ? = 2
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => ? = 1
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] => ? = 3
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] => ? = 2
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => ? = 1
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] => ? = 2
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] => ? = 2
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 0
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 2
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3,2,7,6,5] => ? = 0
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,3,2,6,5,7] => ? = 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,3,2,5,7,6] => ? = 1
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => ? = 0
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => ? = 2
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => ? = 1
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] => ? = 3
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] => ? = 2
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => ? = 1
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] => ? = 2
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] => ? = 2
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => ? = 0
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => ? = 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => ? = 1
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => ? = 0
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => ? = 1
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] => ? = 2
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => ? = 2
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ? = 1
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => ? = 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => ? = 0
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ? = 2
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ? = 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
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: 4% ●values known / values provided: 4%●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: 4% ●values known / values provided: 4%●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
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
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
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 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
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
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
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 0
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 0
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 1
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 1
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
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
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 1
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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