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Your data matches 64 different statistics following compositions of up to 3 maps.
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Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 1 = 2 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 2 = 3 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 1
0000 => 0 = 1 - 1
0001 => 1 = 2 - 1
0010 => 1 = 2 - 1
0011 => 2 = 3 - 1
0100 => 1 = 2 - 1
0101 => 1 = 2 - 1
0110 => 2 = 3 - 1
0111 => 3 = 4 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 2 = 3 - 1
1100 => 2 = 3 - 1
1101 => 2 = 3 - 1
1110 => 3 = 4 - 1
1111 => 4 = 5 - 1
00000 => 0 = 1 - 1
00001 => 1 = 2 - 1
00010 => 1 = 2 - 1
00011 => 2 = 3 - 1
00100 => 1 = 2 - 1
00101 => 1 = 2 - 1
00110 => 2 = 3 - 1
00111 => 3 = 4 - 1
01000 => 1 = 2 - 1
01001 => 1 = 2 - 1
01010 => 1 = 2 - 1
01011 => 2 = 3 - 1
01100 => 2 = 3 - 1
01101 => 2 = 3 - 1
01110 => 3 = 4 - 1
01111 => 4 = 5 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 2 = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000381
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 2
1 => [1,1] => 1
00 => [3] => 3
01 => [2,1] => 2
10 => [1,2] => 2
11 => [1,1,1] => 1
000 => [4] => 4
001 => [3,1] => 3
010 => [2,2] => 2
011 => [2,1,1] => 2
100 => [1,3] => 3
101 => [1,2,1] => 2
110 => [1,1,2] => 2
111 => [1,1,1,1] => 1
0000 => [5] => 5
0001 => [4,1] => 4
0010 => [3,2] => 3
0011 => [3,1,1] => 3
0100 => [2,3] => 3
0101 => [2,2,1] => 2
0110 => [2,1,2] => 2
0111 => [2,1,1,1] => 2
1000 => [1,4] => 4
1001 => [1,3,1] => 3
1010 => [1,2,2] => 2
1011 => [1,2,1,1] => 2
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 2
1110 => [1,1,1,2] => 2
1111 => [1,1,1,1,1] => 1
00000 => [6] => 6
00001 => [5,1] => 5
00010 => [4,2] => 4
00011 => [4,1,1] => 4
00100 => [3,3] => 3
00101 => [3,2,1] => 3
00110 => [3,1,2] => 3
00111 => [3,1,1,1] => 3
01000 => [2,4] => 4
01001 => [2,3,1] => 3
01010 => [2,2,2] => 2
01011 => [2,2,1,1] => 2
01100 => [2,1,3] => 3
01101 => [2,1,2,1] => 2
01110 => [2,1,1,2] => 2
01111 => [2,1,1,1,1] => 2
10000 => [1,5] => 5
10001 => [1,4,1] => 4
10010 => [1,3,2] => 3
10011 => [1,3,1,1] => 3
Description
The largest part of an integer composition.
Matching statistic: St000147
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> 2
1 => [1,1] => [1,1]
=> 1
00 => [3] => [3]
=> 3
01 => [2,1] => [2,1]
=> 2
10 => [1,2] => [2,1]
=> 2
11 => [1,1,1] => [1,1,1]
=> 1
000 => [4] => [4]
=> 4
001 => [3,1] => [3,1]
=> 3
010 => [2,2] => [2,2]
=> 2
011 => [2,1,1] => [2,1,1]
=> 2
100 => [1,3] => [3,1]
=> 3
101 => [1,2,1] => [2,1,1]
=> 2
110 => [1,1,2] => [2,1,1]
=> 2
111 => [1,1,1,1] => [1,1,1,1]
=> 1
0000 => [5] => [5]
=> 5
0001 => [4,1] => [4,1]
=> 4
0010 => [3,2] => [3,2]
=> 3
0011 => [3,1,1] => [3,1,1]
=> 3
0100 => [2,3] => [3,2]
=> 3
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]
=> 2
1000 => [1,4] => [4,1]
=> 4
1001 => [1,3,1] => [3,1,1]
=> 3
1010 => [1,2,2] => [2,2,1]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> 2
1100 => [1,1,3] => [3,1,1]
=> 3
1101 => [1,1,2,1] => [2,1,1,1]
=> 2
1110 => [1,1,1,2] => [2,1,1,1]
=> 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 1
00000 => [6] => [6]
=> 6
00001 => [5,1] => [5,1]
=> 5
00010 => [4,2] => [4,2]
=> 4
00011 => [4,1,1] => [4,1,1]
=> 4
00100 => [3,3] => [3,3]
=> 3
00101 => [3,2,1] => [3,2,1]
=> 3
00110 => [3,1,2] => [3,2,1]
=> 3
00111 => [3,1,1,1] => [3,1,1,1]
=> 3
01000 => [2,4] => [4,2]
=> 4
01001 => [2,3,1] => [3,2,1]
=> 3
01010 => [2,2,2] => [2,2,2]
=> 2
01011 => [2,2,1,1] => [2,2,1,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> 3
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]
=> 2
10000 => [1,5] => [5,1]
=> 5
10001 => [1,4,1] => [4,1,1]
=> 4
10010 => [1,3,2] => [3,2,1]
=> 3
10011 => [1,3,1,1] => [3,1,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000982
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1
1 => [1,1] => 11 => 2
00 => [3] => 100 => 2
01 => [2,1] => 101 => 1
10 => [1,2] => 110 => 2
11 => [1,1,1] => 111 => 3
000 => [4] => 1000 => 3
001 => [3,1] => 1001 => 2
010 => [2,2] => 1010 => 1
011 => [2,1,1] => 1011 => 2
100 => [1,3] => 1100 => 2
101 => [1,2,1] => 1101 => 2
110 => [1,1,2] => 1110 => 3
111 => [1,1,1,1] => 1111 => 4
0000 => [5] => 10000 => 4
0001 => [4,1] => 10001 => 3
0010 => [3,2] => 10010 => 2
0011 => [3,1,1] => 10011 => 2
0100 => [2,3] => 10100 => 2
0101 => [2,2,1] => 10101 => 1
0110 => [2,1,2] => 10110 => 2
0111 => [2,1,1,1] => 10111 => 3
1000 => [1,4] => 11000 => 3
1001 => [1,3,1] => 11001 => 2
1010 => [1,2,2] => 11010 => 2
1011 => [1,2,1,1] => 11011 => 2
1100 => [1,1,3] => 11100 => 3
1101 => [1,1,2,1] => 11101 => 3
1110 => [1,1,1,2] => 11110 => 4
1111 => [1,1,1,1,1] => 11111 => 5
00000 => [6] => 100000 => 5
00001 => [5,1] => 100001 => 4
00010 => [4,2] => 100010 => 3
00011 => [4,1,1] => 100011 => 3
00100 => [3,3] => 100100 => 2
00101 => [3,2,1] => 100101 => 2
00110 => [3,1,2] => 100110 => 2
00111 => [3,1,1,1] => 100111 => 3
01000 => [2,4] => 101000 => 3
01001 => [2,3,1] => 101001 => 2
01010 => [2,2,2] => 101010 => 1
01011 => [2,2,1,1] => 101011 => 2
01100 => [2,1,3] => 101100 => 2
01101 => [2,1,2,1] => 101101 => 2
01110 => [2,1,1,2] => 101110 => 3
01111 => [2,1,1,1,1] => 101111 => 4
10000 => [1,5] => 110000 => 4
10001 => [1,4,1] => 110001 => 3
10010 => [1,3,2] => 110010 => 2
10011 => [1,3,1,1] => 110011 => 2
Description
The length of the longest constant subword.
Matching statistic: St000983
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 2
1 => [1,1] => 11 => 1
00 => [3] => 100 => 2
01 => [2,1] => 101 => 3
10 => [1,2] => 110 => 2
11 => [1,1,1] => 111 => 1
000 => [4] => 1000 => 2
001 => [3,1] => 1001 => 2
010 => [2,2] => 1010 => 4
011 => [2,1,1] => 1011 => 3
100 => [1,3] => 1100 => 2
101 => [1,2,1] => 1101 => 3
110 => [1,1,2] => 1110 => 2
111 => [1,1,1,1] => 1111 => 1
0000 => [5] => 10000 => 2
0001 => [4,1] => 10001 => 2
0010 => [3,2] => 10010 => 3
0011 => [3,1,1] => 10011 => 2
0100 => [2,3] => 10100 => 4
0101 => [2,2,1] => 10101 => 5
0110 => [2,1,2] => 10110 => 3
0111 => [2,1,1,1] => 10111 => 3
1000 => [1,4] => 11000 => 2
1001 => [1,3,1] => 11001 => 2
1010 => [1,2,2] => 11010 => 4
1011 => [1,2,1,1] => 11011 => 3
1100 => [1,1,3] => 11100 => 2
1101 => [1,1,2,1] => 11101 => 3
1110 => [1,1,1,2] => 11110 => 2
1111 => [1,1,1,1,1] => 11111 => 1
00000 => [6] => 100000 => 2
00001 => [5,1] => 100001 => 2
00010 => [4,2] => 100010 => 3
00011 => [4,1,1] => 100011 => 2
00100 => [3,3] => 100100 => 3
00101 => [3,2,1] => 100101 => 4
00110 => [3,1,2] => 100110 => 2
00111 => [3,1,1,1] => 100111 => 2
01000 => [2,4] => 101000 => 4
01001 => [2,3,1] => 101001 => 4
01010 => [2,2,2] => 101010 => 6
01011 => [2,2,1,1] => 101011 => 5
01100 => [2,1,3] => 101100 => 3
01101 => [2,1,2,1] => 101101 => 3
01110 => [2,1,1,2] => 101110 => 3
01111 => [2,1,1,1,1] => 101111 => 3
10000 => [1,5] => 110000 => 2
10001 => [1,4,1] => 110001 => 2
10010 => [1,3,2] => 110010 => 3
10011 => [1,3,1,1] => 110011 => 2
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000010
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [1,1]
=> 2
1 => [1,1] => [1,1]
=> [2]
=> 1
00 => [3] => [3]
=> [1,1,1]
=> 3
01 => [2,1] => [2,1]
=> [2,1]
=> 2
10 => [1,2] => [2,1]
=> [2,1]
=> 2
11 => [1,1,1] => [1,1,1]
=> [3]
=> 1
000 => [4] => [4]
=> [1,1,1,1]
=> 4
001 => [3,1] => [3,1]
=> [2,1,1]
=> 3
010 => [2,2] => [2,2]
=> [2,2]
=> 2
011 => [2,1,1] => [2,1,1]
=> [3,1]
=> 2
100 => [1,3] => [3,1]
=> [2,1,1]
=> 3
101 => [1,2,1] => [2,1,1]
=> [3,1]
=> 2
110 => [1,1,2] => [2,1,1]
=> [3,1]
=> 2
111 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
0000 => [5] => [5]
=> [1,1,1,1,1]
=> 5
0001 => [4,1] => [4,1]
=> [2,1,1,1]
=> 4
0010 => [3,2] => [3,2]
=> [2,2,1]
=> 3
0011 => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 3
0100 => [2,3] => [3,2]
=> [2,2,1]
=> 3
0101 => [2,2,1] => [2,2,1]
=> [3,2]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [3,2]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 2
1000 => [1,4] => [4,1]
=> [2,1,1,1]
=> 4
1001 => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 3
1010 => [1,2,2] => [2,2,1]
=> [3,2]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 2
1100 => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 3
1101 => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 2
1110 => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
00000 => [6] => [6]
=> [1,1,1,1,1,1]
=> 6
00001 => [5,1] => [5,1]
=> [2,1,1,1,1]
=> 5
00010 => [4,2] => [4,2]
=> [2,2,1,1]
=> 4
00011 => [4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 4
00100 => [3,3] => [3,3]
=> [2,2,2]
=> 3
00101 => [3,2,1] => [3,2,1]
=> [3,2,1]
=> 3
00110 => [3,1,2] => [3,2,1]
=> [3,2,1]
=> 3
00111 => [3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> 3
01000 => [2,4] => [4,2]
=> [2,2,1,1]
=> 4
01001 => [2,3,1] => [3,2,1]
=> [3,2,1]
=> 3
01010 => [2,2,2] => [2,2,2]
=> [3,3]
=> 2
01011 => [2,2,1,1] => [2,2,1,1]
=> [4,2]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [3,2,1]
=> 3
01101 => [2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 2
10000 => [1,5] => [5,1]
=> [2,1,1,1,1]
=> 5
10001 => [1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 4
10010 => [1,3,2] => [3,2,1]
=> [3,2,1]
=> 3
10011 => [1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000734
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [[1,2]]
=> 2
1 => [1,1] => [1,1]
=> [[1],[2]]
=> 1
00 => [3] => [3]
=> [[1,2,3]]
=> 3
01 => [2,1] => [2,1]
=> [[1,2],[3]]
=> 2
10 => [1,2] => [2,1]
=> [[1,2],[3]]
=> 2
11 => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 1
000 => [4] => [4]
=> [[1,2,3,4]]
=> 4
001 => [3,1] => [3,1]
=> [[1,2,3],[4]]
=> 3
010 => [2,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
011 => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
100 => [1,3] => [3,1]
=> [[1,2,3],[4]]
=> 3
101 => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
110 => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
111 => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
0000 => [5] => [5]
=> [[1,2,3,4,5]]
=> 5
0001 => [4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
0010 => [3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
0011 => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
0100 => [2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 3
0101 => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
1000 => [1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 4
1001 => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
1010 => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
1100 => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
1101 => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
1110 => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
00000 => [6] => [6]
=> [[1,2,3,4,5,6]]
=> 6
00001 => [5,1] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
00010 => [4,2] => [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
00011 => [4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
00100 => [3,3] => [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
00101 => [3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
00110 => [3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
00111 => [3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
01000 => [2,4] => [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
01001 => [2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
01010 => [2,2,2] => [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
01011 => [2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
01101 => [2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
10000 => [1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
10001 => [1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
10010 => [1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
10011 => [1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000676
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 89%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 89%
Values
0 => [2] => [2]
=> [1,0,1,0]
=> 2
1 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
00 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
01 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
10 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
11 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
010 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
011 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
100 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
101 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
110 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
111 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
0000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
0001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
0010 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
0011 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
0100 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
0101 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
1001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
1010 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1100 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
1101 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1110 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
00000 => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
00001 => [5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
00010 => [4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
00011 => [4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
00100 => [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
00101 => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
00110 => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
00111 => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
01000 => [2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
01001 => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
01010 => [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
01011 => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
01101 => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
10000 => [1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
10001 => [1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
10010 => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
10011 => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
00000000 => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00000001 => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00000011 => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00000111 => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00001111 => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00011111 => [4,1,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00111111 => [3,1,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01011111 => [2,2,1,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01101111 => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01110111 => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111011 => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111101 => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111110 => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111111 => [2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000000 => [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000001 => [1,7,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000011 => [1,6,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000111 => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10001111 => [1,4,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10011111 => [1,3,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10101111 => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10110111 => [1,2,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111011 => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111101 => [1,2,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111111 => [1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000000 => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000001 => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000011 => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000111 => [1,1,4,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11001111 => [1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11010111 => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011011 => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011101 => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011110 => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011111 => [1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100000 => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100001 => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100011 => [1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100111 => [1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101011 => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101101 => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101110 => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101111 => [1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110000 => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110001 => [1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110011 => [1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110101 => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110110 => [1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 89%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 89%
Values
0 => [2] => [2]
=> [1,0,1,0]
=> 2
1 => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
00 => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
01 => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
10 => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
11 => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
000 => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
001 => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
010 => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
011 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
100 => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
101 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
110 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
111 => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
0000 => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
0001 => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
0010 => [3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
0011 => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
0100 => [2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
0101 => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1000 => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
1001 => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
1010 => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1100 => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
1101 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1110 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
00000 => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
00001 => [5,1] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
00010 => [4,2] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
00011 => [4,1,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
00100 => [3,3] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
00101 => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
00110 => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
00111 => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
01000 => [2,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
01001 => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
01010 => [2,2,2] => [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
01011 => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
01101 => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
10000 => [1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
10001 => [1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
10010 => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
10011 => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
00000000 => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00000001 => [8,1] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00000011 => [7,1,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00000111 => [6,1,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00001111 => [5,1,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00011111 => [4,1,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
00111111 => [3,1,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01011111 => [2,2,1,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01101111 => [2,1,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01110111 => [2,1,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111011 => [2,1,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111101 => [2,1,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111110 => [2,1,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
01111111 => [2,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000000 => [1,8] => [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000001 => [1,7,1] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000011 => [1,6,1,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10000111 => [1,5,1,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10001111 => [1,4,1,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10011111 => [1,3,1,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10101111 => [1,2,2,1,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10110111 => [1,2,1,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111011 => [1,2,1,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111101 => [1,2,1,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
10111111 => [1,2,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000000 => [1,1,7] => [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000001 => [1,1,6,1] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000011 => [1,1,5,1,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11000111 => [1,1,4,1,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11001111 => [1,1,3,1,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11010111 => [1,1,2,2,1,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011011 => [1,1,2,1,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011101 => [1,1,2,1,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011110 => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11011111 => [1,1,2,1,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100000 => [1,1,1,6] => [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100001 => [1,1,1,5,1] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100011 => [1,1,1,4,1,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11100111 => [1,1,1,3,1,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101011 => [1,1,1,2,2,1,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101101 => [1,1,1,2,1,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101110 => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11101111 => [1,1,1,2,1,1,1,1] => [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110000 => [1,1,1,1,5] => [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110001 => [1,1,1,1,4,1] => [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110011 => [1,1,1,1,3,1,1] => [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110101 => [1,1,1,1,2,2,1] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
11110110 => [1,1,1,1,2,1,2] => [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,9}
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001291
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 68%●distinct values known / distinct values provided: 56%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 68%●distinct values known / distinct values provided: 56%
Values
0 => [2] => [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
1 => [1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
00 => [3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
01 => [2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
10 => [1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
11 => [1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
000 => [4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
001 => [3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
010 => [2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
011 => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
100 => [1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
101 => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
110 => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
111 => [1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
0000 => [5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
0001 => [4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
0010 => [3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
0011 => [3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
0100 => [2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
0101 => [2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
0110 => [2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
0111 => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
1000 => [1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
1001 => [1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
1010 => [1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
1011 => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
1100 => [1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
1101 => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
1110 => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
00000 => [6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,6} + 1
00001 => [5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
00010 => [4,2] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
00011 => [4,1,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
00100 => [3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
00101 => [3,2,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
00110 => [3,1,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
00111 => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
01000 => [2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
01001 => [2,3,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
01010 => [2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
01011 => [2,2,1,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
01100 => [2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
01101 => [2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
01110 => [2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
10000 => [1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
10001 => [1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
10010 => [1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
10011 => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
10100 => [1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,6} + 1
000000 => [7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
000001 => [6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
100000 => [1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
101111 => [1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
110111 => [1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
111011 => [1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
111101 => [1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,6,6,7} + 1
0000000 => [8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0000001 => [7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0000010 => [6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0000011 => [6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0011111 => [3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0100000 => [2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0101111 => [2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0110111 => [2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0111011 => [2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0111101 => [2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0111110 => [2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
0111111 => [2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1000000 => [1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1000001 => [1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1001111 => [1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1010111 => [1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1011011 => [1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1011101 => [1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1011111 => [1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1100000 => [1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1100111 => [1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1101011 => [1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1101101 => [1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1101111 => [1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1110011 => [1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1110101 => [1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1110110 => [1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1110111 => [1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111001 => [1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111010 => [1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111011 => [1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111100 => [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111101 => [1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,6,6,6,6,6,7,7,8} + 1
00000000 => [9] => [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,8,8,9} + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000013The height of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001645The pebbling number of a connected graph. St000439The position of the first down step of a Dyck path. St000306The bounce count of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000662The staircase size of the code of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000141The maximum drop size of a permutation. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000686The finitistic dominant dimension of a Dyck path. St000025The number of initial rises of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000028The number of stack-sorts needed to sort a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000308The height of the tree associated to a permutation. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001652The length of a longest interval of consecutive numbers. St000454The largest eigenvalue of a graph if it is integral. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000094The depth of an ordered tree. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral.
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