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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St000374
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
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000374: 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] => 1
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
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] => 1
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] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
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] => 1
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] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
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] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000703
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
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000703: 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] => 1
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
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] => 1
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] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
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] => 1
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] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
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] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 2
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000994
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
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000994: 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] => 1
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
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] => 1
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] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
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] => 1
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] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
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] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 2
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St001280
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,1]
=> 0
1 => 0 => [2] => [2]
=> 1
00 => 11 => [1,1,1] => [1,1,1]
=> 0
01 => 10 => [1,2] => [2,1]
=> 1
10 => 01 => [2,1] => [2,1]
=> 1
11 => 00 => [3] => [3]
=> 1
000 => 111 => [1,1,1,1] => [1,1,1,1]
=> 0
001 => 110 => [1,1,2] => [2,1,1]
=> 1
010 => 101 => [1,2,1] => [2,1,1]
=> 1
011 => 100 => [1,3] => [3,1]
=> 1
100 => 011 => [2,1,1] => [2,1,1]
=> 1
101 => 010 => [2,2] => [2,2]
=> 2
110 => 001 => [3,1] => [3,1]
=> 1
111 => 000 => [4] => [4]
=> 1
0000 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
0001 => 1110 => [1,1,1,2] => [2,1,1,1]
=> 1
0010 => 1101 => [1,1,2,1] => [2,1,1,1]
=> 1
0011 => 1100 => [1,1,3] => [3,1,1]
=> 1
0100 => 1011 => [1,2,1,1] => [2,1,1,1]
=> 1
0101 => 1010 => [1,2,2] => [2,2,1]
=> 2
0110 => 1001 => [1,3,1] => [3,1,1]
=> 1
0111 => 1000 => [1,4] => [4,1]
=> 1
1000 => 0111 => [2,1,1,1] => [2,1,1,1]
=> 1
1001 => 0110 => [2,1,2] => [2,2,1]
=> 2
1010 => 0101 => [2,2,1] => [2,2,1]
=> 2
1011 => 0100 => [2,3] => [3,2]
=> 2
1100 => 0011 => [3,1,1] => [3,1,1]
=> 1
1101 => 0010 => [3,2] => [3,2]
=> 2
1110 => 0001 => [4,1] => [4,1]
=> 1
1111 => 0000 => [5] => [5]
=> 1
00000 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
00001 => 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 1
00010 => 11101 => [1,1,1,2,1] => [2,1,1,1,1]
=> 1
00011 => 11100 => [1,1,1,3] => [3,1,1,1]
=> 1
00100 => 11011 => [1,1,2,1,1] => [2,1,1,1,1]
=> 1
00101 => 11010 => [1,1,2,2] => [2,2,1,1]
=> 2
00110 => 11001 => [1,1,3,1] => [3,1,1,1]
=> 1
00111 => 11000 => [1,1,4] => [4,1,1]
=> 1
01000 => 10111 => [1,2,1,1,1] => [2,1,1,1,1]
=> 1
01001 => 10110 => [1,2,1,2] => [2,2,1,1]
=> 2
01010 => 10101 => [1,2,2,1] => [2,2,1,1]
=> 2
01011 => 10100 => [1,2,3] => [3,2,1]
=> 2
01100 => 10011 => [1,3,1,1] => [3,1,1,1]
=> 1
01101 => 10010 => [1,3,2] => [3,2,1]
=> 2
01110 => 10001 => [1,4,1] => [4,1,1]
=> 1
01111 => 10000 => [1,5] => [5,1]
=> 1
10000 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1
10001 => 01110 => [2,1,1,2] => [2,2,1,1]
=> 2
10010 => 01101 => [2,1,2,1] => [2,2,1,1]
=> 2
10011 => 01100 => [2,1,3] => [3,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000390
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000390: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 1
00 => 0
01 => 1
10 => 1
11 => 1
000 => 0
001 => 1
010 => 1
011 => 1
100 => 1
101 => 2
110 => 1
111 => 1
0000 => 0
0001 => 1
0010 => 1
0011 => 1
0100 => 1
0101 => 2
0110 => 1
0111 => 1
1000 => 1
1001 => 2
1010 => 2
1011 => 2
1100 => 1
1101 => 2
1110 => 1
1111 => 1
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 => 1
10001 => 2
10010 => 2
10011 => 2
=> ? = 0
Description
The number of runs of ones in a binary word.
Matching statistic: St000884
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
St000884: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Values
0 => [2] => [1,1,0,0]
=> [1,2] => 0
1 => [1,1] => [1,0,1,0]
=> [2,1] => 1
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
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] => 1
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] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
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] => 1
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] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
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] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
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
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 2
10101010101 => [1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 6
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => 11 => 0
1 => [1,1] => [2] => 10 => 1
00 => [3] => [1,1,1] => 111 => 0
01 => [2,1] => [2,1] => 101 => 1
10 => [1,2] => [1,2] => 110 => 1
11 => [1,1,1] => [3] => 100 => 1
000 => [4] => [1,1,1,1] => 1111 => 0
001 => [3,1] => [2,1,1] => 1011 => 1
010 => [2,2] => [1,2,1] => 1101 => 1
011 => [2,1,1] => [3,1] => 1001 => 1
100 => [1,3] => [1,1,2] => 1110 => 1
101 => [1,2,1] => [2,2] => 1010 => 2
110 => [1,1,2] => [1,3] => 1100 => 1
111 => [1,1,1,1] => [4] => 1000 => 1
0000 => [5] => [1,1,1,1,1] => 11111 => 0
0001 => [4,1] => [2,1,1,1] => 10111 => 1
0010 => [3,2] => [1,2,1,1] => 11011 => 1
0011 => [3,1,1] => [3,1,1] => 10011 => 1
0100 => [2,3] => [1,1,2,1] => 11101 => 1
0101 => [2,2,1] => [2,2,1] => 10101 => 2
0110 => [2,1,2] => [1,3,1] => 11001 => 1
0111 => [2,1,1,1] => [4,1] => 10001 => 1
1000 => [1,4] => [1,1,1,2] => 11110 => 1
1001 => [1,3,1] => [2,1,2] => 10110 => 2
1010 => [1,2,2] => [1,2,2] => 11010 => 2
1011 => [1,2,1,1] => [3,2] => 10010 => 2
1100 => [1,1,3] => [1,1,3] => 11100 => 1
1101 => [1,1,2,1] => [2,3] => 10100 => 2
1110 => [1,1,1,2] => [1,4] => 11000 => 1
1111 => [1,1,1,1,1] => [5] => 10000 => 1
00000 => [6] => [1,1,1,1,1,1] => 111111 => 0
00001 => [5,1] => [2,1,1,1,1] => 101111 => 1
00010 => [4,2] => [1,2,1,1,1] => 110111 => 1
00011 => [4,1,1] => [3,1,1,1] => 100111 => 1
00100 => [3,3] => [1,1,2,1,1] => 111011 => 1
00101 => [3,2,1] => [2,2,1,1] => 101011 => 2
00110 => [3,1,2] => [1,3,1,1] => 110011 => 1
00111 => [3,1,1,1] => [4,1,1] => 100011 => 1
01000 => [2,4] => [1,1,1,2,1] => 111101 => 1
01001 => [2,3,1] => [2,1,2,1] => 101101 => 2
01010 => [2,2,2] => [1,2,2,1] => 110101 => 2
01011 => [2,2,1,1] => [3,2,1] => 100101 => 2
01100 => [2,1,3] => [1,1,3,1] => 111001 => 1
01101 => [2,1,2,1] => [2,3,1] => 101001 => 2
01110 => [2,1,1,2] => [1,4,1] => 110001 => 1
01111 => [2,1,1,1,1] => [5,1] => 100001 => 1
10000 => [1,5] => [1,1,1,1,2] => 111110 => 1
10001 => [1,4,1] => [2,1,1,2] => 101110 => 2
10010 => [1,3,2] => [1,2,1,2] => 110110 => 2
10011 => [1,3,1,1] => [3,1,2] => 100110 => 2
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 0
000000001 => [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
111111110 => [1,1,1,1,1,1,1,1,2] => [1,9] => 1100000000 => ? = 1
0000000001 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
Description
The number of descents of a binary word.
Matching statistic: St000659
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 86%●distinct values known / distinct values provided: 71%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 86%●distinct values known / distinct values provided: 71%
Values
0 => [2] => [1,1] => [1,0,1,0]
=> 0
1 => [1,1] => [2] => [1,1,0,0]
=> 1
00 => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
01 => [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
10 => [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
010 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
100 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
101 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
0000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
0010 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
0011 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
0100 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
1000 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
1001 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
1011 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
1100 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
00000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
00010 => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
00011 => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
00100 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
00101 => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
00110 => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
00111 => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
01000 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
01001 => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
01100 => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
10000 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
10001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
10010 => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
10011 => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2
0001111 => [4,1,1,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
1100011 => [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
1110111 => [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
1111111 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00000001 => [8,1] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
10000000 => [1,8] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
11111100 => [1,1,1,1,1,1,3] => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
11111110 => [1,1,1,1,1,1,1,2] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
11111111 => [1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
000000001 => [9,1] => [2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
100000000 => [1,9] => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
101010101 => [1,2,2,2,2,1] => [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
111111110 => [1,1,1,1,1,1,1,1,2] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
1000000000 => [1,10] => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
=> [1] => [1] => [1,0]
=> ? = 0
0000000001 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ?
=> ? = 1
10101010101 => [1,2,2,2,2,2,1] => [2,2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000658
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 84%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 84%●distinct values known / distinct values provided: 71%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 2
1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
000000001 => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
101010101 => [1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
=> [1] => [1,0]
=> [1,0]
=> ? = 0
0000000001 => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
10101010101 => [1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000340
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 84%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 84%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 2
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
000000001 => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
101010101 => [1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 5
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
0000000001 => [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
10101010101 => [1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000035The number of left outer peaks of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000834The number of right outer peaks of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001512The minimum rank of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances 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. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St001874Lusztig's a-function for the symmetric group. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation.
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