Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001556
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,2] => 0
1 => [1,1] => [1,0,1,0]
 => [2,1] => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 1
Description
The number of inversions of the third entry of a permutation. This is, for a permutation $\pi$ of length $n$, $$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001722
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
0 => [2] => [1,1,0,0]
 => 1100 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => 1010 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => 111000 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => 110010 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => 11110000 => ? = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 1 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 1 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => ? = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => ? = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => ? = 1 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 0 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? = 0 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 1 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000782
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
0 => [2] => [1,1,0,0]
 => [(1,4),(2,3)]
 => ? = 0 + 1
1 => [1,1] => [1,0,1,0]
 => [(1,2),(3,4)]
 => ? = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => ? = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => ? = 1 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => ? = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => ? = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => ? = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => ? = 1 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => ? = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => ? = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => ? = 1 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => ? = 0 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => ? = 0 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => ? = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => ? = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => ? = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => ? = 1 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => ? = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => ? = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => ? = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => ? = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => ? = 1 + 1
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001605
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => []
 => ? = 0 + 1
1 => [1,1] => [[1,1],[]]
 => []
 => ? = 0 + 1
00 => [3] => [[3],[]]
 => []
 => ? = 0 + 1
01 => [2,1] => [[2,2],[1]]
 => [1]
 => ? = 0 + 1
10 => [1,2] => [[2,1],[]]
 => []
 => ? = 0 + 1
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? = 0 + 1
000 => [4] => [[4],[]]
 => []
 => ? = 0 + 1
001 => [3,1] => [[3,3],[2]]
 => [2]
 => ? = 1 + 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => ? = 0 + 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => ? = 1 + 1
100 => [1,3] => [[3,1],[]]
 => []
 => ? = 0 + 1
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => ? = 1 + 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? = 0 + 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? = 1 + 1
0000 => [5] => [[5],[]]
 => []
 => ? = 0 + 1
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1 = 0 + 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => ? = 1 + 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2 = 1 + 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 0 + 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1 = 0 + 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => ? = 1 + 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 2 = 1 + 1
1000 => [1,4] => [[4,1],[]]
 => []
 => ? = 0 + 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => ? = 0 + 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? = 1 + 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => ? = 1 + 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? = 0 + 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => ? = 0 + 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? = 1 + 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? = 1 + 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.