Your data matches 88 different statistics following compositions of up to 3 maps.
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St000291: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 0
100 => 1
101 => 1
110 => 1
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 1
0101 => 1
0110 => 1
0111 => 0
1000 => 1
1001 => 1
1010 => 2
1011 => 1
1100 => 1
1101 => 1
1110 => 1
1111 => 0
00000 => 0
00001 => 0
00010 => 1
00011 => 0
00100 => 1
00101 => 1
00110 => 1
00111 => 0
01000 => 1
01001 => 1
01010 => 2
01011 => 1
01100 => 1
01101 => 1
01110 => 1
01111 => 0
10000 => 1
10001 => 1
10010 => 2
10011 => 1
Description
The number of descents of a binary word.
St000292: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 1
10 => 0
11 => 0
000 => 0
001 => 1
010 => 1
011 => 1
100 => 0
101 => 1
110 => 0
111 => 0
0000 => 0
0001 => 1
0010 => 1
0011 => 1
0100 => 1
0101 => 2
0110 => 1
0111 => 1
1000 => 0
1001 => 1
1010 => 1
1011 => 1
1100 => 0
1101 => 1
1110 => 0
1111 => 0
00000 => 0
00001 => 1
00010 => 1
00011 => 1
00100 => 1
00101 => 2
00110 => 1
00111 => 1
01000 => 1
01001 => 2
01010 => 2
01011 => 2
01100 => 1
01101 => 2
01110 => 1
01111 => 1
10000 => 0
10001 => 1
10010 => 1
10011 => 1
Description
The number of ascents of a binary word.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The number of factors DDU in a Dyck path.
Mp00224: Binary words runsortBinary words
Mp00268: Binary words zeros to flag zerosBinary words
St000628: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 10 => 1
01 => 01 => 00 => 0
10 => 01 => 00 => 0
11 => 11 => 11 => 0
000 => 000 => 010 => 1
001 => 001 => 110 => 1
010 => 001 => 110 => 1
011 => 011 => 000 => 0
100 => 001 => 110 => 1
101 => 011 => 000 => 0
110 => 011 => 000 => 0
111 => 111 => 111 => 0
0000 => 0000 => 1010 => 1
0001 => 0001 => 0010 => 1
0010 => 0001 => 0010 => 1
0011 => 0011 => 1110 => 1
0100 => 0001 => 0010 => 1
0101 => 0101 => 1100 => 2
0110 => 0011 => 1110 => 1
0111 => 0111 => 0000 => 0
1000 => 0001 => 0010 => 1
1001 => 0011 => 1110 => 1
1010 => 0011 => 1110 => 1
1011 => 0111 => 0000 => 0
1100 => 0011 => 1110 => 1
1101 => 0111 => 0000 => 0
1110 => 0111 => 0000 => 0
1111 => 1111 => 1111 => 0
00000 => 00000 => 01010 => 1
00001 => 00001 => 11010 => 1
00010 => 00001 => 11010 => 1
00011 => 00011 => 00010 => 1
00100 => 00001 => 11010 => 1
00101 => 00101 => 00110 => 2
00110 => 00011 => 00010 => 1
00111 => 00111 => 11110 => 1
01000 => 00001 => 11010 => 1
01001 => 00101 => 00110 => 2
01010 => 00101 => 00110 => 2
01011 => 01011 => 11100 => 2
01100 => 00011 => 00010 => 1
01101 => 01011 => 11100 => 2
01110 => 00111 => 11110 => 1
01111 => 01111 => 00000 => 0
10000 => 00001 => 11010 => 1
10001 => 00011 => 00010 => 1
10010 => 00011 => 00010 => 1
10011 => 00111 => 11110 => 1
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Mp00261: Binary words Burrows-WheelerBinary words
Mp00268: Binary words zeros to flag zerosBinary words
St000875: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 10 => 1
01 => 10 => 01 => 0
10 => 10 => 01 => 0
11 => 11 => 11 => 0
000 => 000 => 010 => 1
001 => 100 => 101 => 1
010 => 100 => 101 => 1
011 => 110 => 011 => 0
100 => 100 => 101 => 1
101 => 110 => 011 => 0
110 => 110 => 011 => 0
111 => 111 => 111 => 0
0000 => 0000 => 1010 => 2
0001 => 1000 => 0101 => 1
0010 => 1000 => 0101 => 1
0011 => 1010 => 1001 => 1
0100 => 1000 => 0101 => 1
0101 => 1100 => 1011 => 1
0110 => 1010 => 1001 => 1
0111 => 1110 => 0111 => 0
1000 => 1000 => 0101 => 1
1001 => 1010 => 1001 => 1
1010 => 1100 => 1011 => 1
1011 => 1110 => 0111 => 0
1100 => 1010 => 1001 => 1
1101 => 1110 => 0111 => 0
1110 => 1110 => 0111 => 0
1111 => 1111 => 1111 => 0
00000 => 00000 => 01010 => 2
00001 => 10000 => 10101 => 2
00010 => 10000 => 10101 => 2
00011 => 10010 => 01101 => 1
00100 => 10000 => 10101 => 2
00101 => 11000 => 01011 => 1
00110 => 10010 => 01101 => 1
00111 => 10110 => 10001 => 1
01000 => 10000 => 10101 => 2
01001 => 11000 => 01011 => 1
01010 => 11000 => 01011 => 1
01011 => 11100 => 10111 => 1
01100 => 10010 => 01101 => 1
01101 => 11100 => 10111 => 1
01110 => 10110 => 10001 => 1
01111 => 11110 => 01111 => 0
10000 => 10000 => 10101 => 2
10001 => 10010 => 01101 => 1
10010 => 11000 => 01011 => 1
10011 => 10110 => 10001 => 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Mp00261: Binary words Burrows-WheelerBinary words
Mp00268: Binary words zeros to flag zerosBinary words
St001421: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 10 => 1
01 => 10 => 01 => 0
10 => 10 => 01 => 0
11 => 11 => 11 => 0
000 => 000 => 010 => 1
001 => 100 => 101 => 1
010 => 100 => 101 => 1
011 => 110 => 011 => 0
100 => 100 => 101 => 1
101 => 110 => 011 => 0
110 => 110 => 011 => 0
111 => 111 => 111 => 0
0000 => 0000 => 1010 => 2
0001 => 1000 => 0101 => 1
0010 => 1000 => 0101 => 1
0011 => 1010 => 1001 => 1
0100 => 1000 => 0101 => 1
0101 => 1100 => 1011 => 1
0110 => 1010 => 1001 => 1
0111 => 1110 => 0111 => 0
1000 => 1000 => 0101 => 1
1001 => 1010 => 1001 => 1
1010 => 1100 => 1011 => 1
1011 => 1110 => 0111 => 0
1100 => 1010 => 1001 => 1
1101 => 1110 => 0111 => 0
1110 => 1110 => 0111 => 0
1111 => 1111 => 1111 => 0
00000 => 00000 => 01010 => 2
00001 => 10000 => 10101 => 2
00010 => 10000 => 10101 => 2
00011 => 10010 => 01101 => 1
00100 => 10000 => 10101 => 2
00101 => 11000 => 01011 => 1
00110 => 10010 => 01101 => 1
00111 => 10110 => 10001 => 1
01000 => 10000 => 10101 => 2
01001 => 11000 => 01011 => 1
01010 => 11000 => 01011 => 1
01011 => 11100 => 10111 => 1
01100 => 10010 => 01101 => 1
01101 => 11100 => 10111 => 1
01110 => 10110 => 10001 => 1
01111 => 11110 => 01111 => 0
10000 => 10000 => 10101 => 2
10001 => 10010 => 01101 => 1
10010 => 11000 => 01011 => 1
10011 => 10110 => 10001 => 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Mp00178: Binary words to compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000390: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 1 = 0 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 2 = 1 + 1
10 => [1,2] => 110 => 1 = 0 + 1
11 => [1,1,1] => 111 => 1 = 0 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 2 = 1 + 1
010 => [2,2] => 1010 => 2 = 1 + 1
011 => [2,1,1] => 1011 => 2 = 1 + 1
100 => [1,3] => 1100 => 1 = 0 + 1
101 => [1,2,1] => 1101 => 2 = 1 + 1
110 => [1,1,2] => 1110 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 1 = 0 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 2 = 1 + 1
0010 => [3,2] => 10010 => 2 = 1 + 1
0011 => [3,1,1] => 10011 => 2 = 1 + 1
0100 => [2,3] => 10100 => 2 = 1 + 1
0101 => [2,2,1] => 10101 => 3 = 2 + 1
0110 => [2,1,2] => 10110 => 2 = 1 + 1
0111 => [2,1,1,1] => 10111 => 2 = 1 + 1
1000 => [1,4] => 11000 => 1 = 0 + 1
1001 => [1,3,1] => 11001 => 2 = 1 + 1
1010 => [1,2,2] => 11010 => 2 = 1 + 1
1011 => [1,2,1,1] => 11011 => 2 = 1 + 1
1100 => [1,1,3] => 11100 => 1 = 0 + 1
1101 => [1,1,2,1] => 11101 => 2 = 1 + 1
1110 => [1,1,1,2] => 11110 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 1 = 0 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 2 = 1 + 1
00010 => [4,2] => 100010 => 2 = 1 + 1
00011 => [4,1,1] => 100011 => 2 = 1 + 1
00100 => [3,3] => 100100 => 2 = 1 + 1
00101 => [3,2,1] => 100101 => 3 = 2 + 1
00110 => [3,1,2] => 100110 => 2 = 1 + 1
00111 => [3,1,1,1] => 100111 => 2 = 1 + 1
01000 => [2,4] => 101000 => 2 = 1 + 1
01001 => [2,3,1] => 101001 => 3 = 2 + 1
01010 => [2,2,2] => 101010 => 3 = 2 + 1
01011 => [2,2,1,1] => 101011 => 3 = 2 + 1
01100 => [2,1,3] => 101100 => 2 = 1 + 1
01101 => [2,1,2,1] => 101101 => 3 = 2 + 1
01110 => [2,1,1,2] => 101110 => 2 = 1 + 1
01111 => [2,1,1,1,1] => 101111 => 2 = 1 + 1
10000 => [1,5] => 110000 => 1 = 0 + 1
10001 => [1,4,1] => 110001 => 2 = 1 + 1
10010 => [1,3,2] => 110010 => 2 = 1 + 1
10011 => [1,3,1,1] => 110011 => 2 = 1 + 1
Description
The number of runs of ones in a binary word.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St000023: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
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] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 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] => 0
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] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
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] => 1
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] => 0
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] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 1
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] => 1
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] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 1
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] => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1
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] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 1
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> []
=> 0
1 => [1,1] => [[1,1],[]]
=> []
=> 0
00 => [3] => [[3],[]]
=> []
=> 0
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
10 => [1,2] => [[2,1],[]]
=> []
=> 0
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 0
000 => [4] => [[4],[]]
=> []
=> 0
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
100 => [1,3] => [[3,1],[]]
=> []
=> 0
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0
0000 => [5] => [[5],[]]
=> []
=> 0
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
1000 => [1,4] => [[4,1],[]]
=> []
=> 0
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0
00000 => [6] => [[6],[]]
=> []
=> 0
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1
10000 => [1,5] => [[5,1],[]]
=> []
=> 0
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000183The side length of the Durfee square of an integer partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000257The number of distinct parts of a partition that occur at least twice. St000293The number of inversions of a binary word. St000353The number of inner valleys of a permutation. St000356The number of occurrences of the pattern 13-2. St000691The number of changes of a binary word. St000779The tier of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000201The number of leaf nodes in a binary tree. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001354The number of series nodes in the modular decomposition of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000260The radius of a connected graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000256The number of parts from which one can substract 2 and still get an integer partition. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001487The number of inner corners of a skew partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.