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Your data matches 458 different statistics following compositions of up to 3 maps.
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Matching statistic: St000629
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000629: 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 => 0
11 => 0
000 => 0
001 => 0
010 => 0
011 => 0
100 => 0
101 => 0
110 => 0
111 => 0
0000 => 0
0001 => 0
0010 => 0
0011 => 0
0100 => 0
0101 => 0
0110 => 0
0111 => 0
1000 => 0
1001 => 0
1010 => 0
1011 => 0
1100 => 0
1101 => 0
1110 => 0
1111 => 0
00000 => 0
00001 => 0
00010 => 0
00011 => 0
00100 => 0
00101 => 0
00110 => 0
00111 => 0
01000 => 0
01001 => 0
01010 => 0
01011 => 0
01100 => 0
01101 => 0
01110 => 0
01111 => 0
10000 => 0
10001 => 0
10010 => 0
10011 => 0
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St001786
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001786: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 22%●distinct values known / distinct values provided: 20%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001786: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 22%●distinct values known / distinct values provided: 20%
Values
0 => [1] => [1,0]
=> 1 = 0 + 1
1 => [1] => [1,0]
=> 1 = 0 + 1
00 => [2] => [1,1,0,0]
=> 1 = 0 + 1
01 => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
10 => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
11 => [2] => [1,1,0,0]
=> 1 = 0 + 1
000 => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
011 => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
100 => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
110 => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
111 => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
00000110 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
00001001 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
00001011 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
00001101 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 1
00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 1
00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0 + 1
00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0 + 1
00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
00100001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0 + 1
00100010 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
00100011 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0 + 1
00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
00100101 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
00100110 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
00100111 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
00101100 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
00101110 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
00101111 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
00110001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
00110010 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 1
00110100 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 1
00110110 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
00110111 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
00111000 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
00111001 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
00111011 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
00111101 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
01000010 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0 + 1
01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0 + 1
01000110 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
01000111 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
01001000 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 0 + 1
01001001 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0 + 1
01001101 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0 + 1
01001110 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0 + 1
01001111 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
01011000 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0 + 1
01011001 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
01011101 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
Description
The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.
Alternatively, remark that the monomials of the polynomial $\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with $n$ east steps: the exponent of $z_i$ is the number of north steps before the $i$-th east step, see [2]. Thus, this statistic records the coefficients of the monomials.
A formula for the coefficient of $z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]:
$$
c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}.
$$
This polynomial arises in a partial symmetrization process as follows, see [1]. For $w\in\frak{S}_n$, let $w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$. Furthermore, let
$$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$$
Then $\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$.
Matching statistic: St000386
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 22%●distinct values known / distinct values provided: 20%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 22%●distinct values known / distinct values provided: 20%
Values
0 => [1] => [1,0]
=> [1,0]
=> 0
1 => [1] => [1,0]
=> [1,0]
=> 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
011 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
100 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
00010100 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0
00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 0
00100011 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
00101100 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
00110011 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
00110100 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
00110110 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
00111000 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
00111001 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
00111011 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0
01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
01000110 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
01001001 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
01001011 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
01001100 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
01010011 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
01011001 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
01011011 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
01011100 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
01100011 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 0
01100100 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
01100110 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
01101001 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
01101011 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
01101100 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
01110001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 0
01110010 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
01110100 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
01110110 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
01110111 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
10001001 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
10001011 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
10001101 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
10001110 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 0
10010011 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
10010100 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
10010110 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
10011001 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
10011011 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001172
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
0 => 0 => [1] => [1,0]
=> 0
1 => 1 => [1] => [1,0]
=> 0
00 => 00 => [2] => [1,1,0,0]
=> 0
01 => 10 => [1,1] => [1,0,1,0]
=> 0
10 => 01 => [1,1] => [1,0,1,0]
=> 0
11 => 11 => [2] => [1,1,0,0]
=> 0
000 => 000 => [3] => [1,1,1,0,0,0]
=> 0
001 => 100 => [1,2] => [1,0,1,1,0,0]
=> 0
010 => 001 => [2,1] => [1,1,0,0,1,0]
=> 0
011 => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
100 => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
101 => 110 => [2,1] => [1,1,0,0,1,0]
=> 0
110 => 011 => [1,2] => [1,0,1,1,0,0]
=> 0
111 => 111 => [3] => [1,1,1,0,0,0]
=> 0
0000 => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
0001 => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
0010 => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
0011 => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
0100 => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
0101 => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0110 => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
0111 => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1000 => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1001 => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
1010 => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
1011 => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
1100 => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
1101 => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
1110 => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
1111 => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 0
00000 => 00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
00001 => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
00010 => 00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
00011 => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
00100 => 00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
00101 => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
00110 => 00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
00111 => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
01000 => 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
01001 => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
01010 => 00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
01011 => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
01100 => 00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
01101 => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
01110 => 00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
01111 => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
10000 => 01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
10001 => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
10010 => 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
10011 => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
00000000 => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
00000010 => 00000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
00000100 => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
00000110 => 00000011 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
00001000 => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 0
00001010 => 00000101 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
00001100 => 00000110 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
00001110 => 00000111 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
00010000 => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
00010010 => 00001001 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
00010100 => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
00010110 => 00001011 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
00011000 => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
00011010 => 00001101 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
00011100 => 00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
00011110 => 00001111 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
00100000 => 00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
00100010 => 00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
00100100 => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
00100110 => 00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
00101000 => 00010100 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
00101010 => 00010101 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00101100 => 00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
00101110 => 00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
00110000 => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
00110010 => 00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
00110100 => 00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
00110110 => 00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
00111000 => 00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
00111010 => 00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
00111100 => 00011110 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
00111110 => 00011111 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
11000001 => 11100000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
11000011 => 11100001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
11000101 => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
11000111 => 11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
11001001 => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
11001011 => 11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
11001101 => 11100110 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
11001111 => 11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
11010001 => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
11010011 => 11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
11010101 => 11101010 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
11010111 => 11101011 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
11011001 => 11101100 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
11011011 => 11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
11011101 => 11101110 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
11011111 => 11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
11100001 => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
11100011 => 11110001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001584
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001584: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
0 => 0 => [1] => [1,0]
=> 0
1 => 1 => [1] => [1,0]
=> 0
00 => 00 => [2] => [1,1,0,0]
=> 0
01 => 10 => [1,1] => [1,0,1,0]
=> 0
10 => 01 => [1,1] => [1,0,1,0]
=> 0
11 => 11 => [2] => [1,1,0,0]
=> 0
000 => 000 => [3] => [1,1,1,0,0,0]
=> 0
001 => 100 => [1,2] => [1,0,1,1,0,0]
=> 0
010 => 001 => [2,1] => [1,1,0,0,1,0]
=> 0
011 => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
100 => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
101 => 110 => [2,1] => [1,1,0,0,1,0]
=> 0
110 => 011 => [1,2] => [1,0,1,1,0,0]
=> 0
111 => 111 => [3] => [1,1,1,0,0,0]
=> 0
0000 => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
0001 => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
0010 => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
0011 => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
0100 => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
0101 => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0110 => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
0111 => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1000 => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1001 => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
1010 => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
1011 => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
1100 => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
1101 => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
1110 => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
1111 => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 0
00000 => 00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
00001 => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
00010 => 00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
00011 => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
00100 => 00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
00101 => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
00110 => 00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
00111 => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
01000 => 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
01001 => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
01010 => 00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
01011 => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
01100 => 00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
01101 => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
01110 => 00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
01111 => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
10000 => 01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
10001 => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
10010 => 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
10011 => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
00000000 => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
00000010 => 00000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
00000100 => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
00000110 => 00000011 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
00001000 => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 0
00001010 => 00000101 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
00001100 => 00000110 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
00001110 => 00000111 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
00010000 => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
00010010 => 00001001 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
00010100 => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
00010110 => 00001011 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
00011000 => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
00011010 => 00001101 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
00011100 => 00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
00011110 => 00001111 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
00100000 => 00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
00100010 => 00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
00100100 => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
00100110 => 00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
00101000 => 00010100 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
00101010 => 00010101 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00101100 => 00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
00101110 => 00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
00110000 => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
00110010 => 00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
00110100 => 00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
00110110 => 00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
00111000 => 00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
00111010 => 00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
00111100 => 00011110 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
00111110 => 00011111 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
11000001 => 11100000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
11000011 => 11100001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
11000101 => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
11000111 => 11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
11001001 => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
11001011 => 11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
11001101 => 11100110 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
11001111 => 11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
11010001 => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
11010011 => 11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
11010101 => 11101010 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
11010111 => 11101011 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
11011001 => 11101100 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
11011011 => 11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
11011101 => 11101110 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
11011111 => 11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
11100001 => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
11100011 => 11110001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
Description
The area statistic between a Dyck path and its bounce path.
The bounce path [[Mp00099]] is weakly below a given Dyck path and this statistic records the number of boxes between the two paths.
Matching statistic: St000980
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
0 => 0 => [1] => [1,0]
=> ? = 0
1 => 1 => [1] => [1,0]
=> ? = 0
00 => 00 => [2] => [1,1,0,0]
=> 0
01 => 10 => [1,1] => [1,0,1,0]
=> 0
10 => 01 => [1,1] => [1,0,1,0]
=> 0
11 => 11 => [2] => [1,1,0,0]
=> 0
000 => 000 => [3] => [1,1,1,0,0,0]
=> 0
001 => 100 => [1,2] => [1,0,1,1,0,0]
=> 0
010 => 001 => [2,1] => [1,1,0,0,1,0]
=> 0
011 => 101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
100 => 010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
101 => 110 => [2,1] => [1,1,0,0,1,0]
=> 0
110 => 011 => [1,2] => [1,0,1,1,0,0]
=> 0
111 => 111 => [3] => [1,1,1,0,0,0]
=> 0
0000 => 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
0001 => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
0010 => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
0011 => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
0100 => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
0101 => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0110 => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
0111 => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1000 => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1001 => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
1010 => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
1011 => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
1100 => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
1101 => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
1110 => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
1111 => 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 0
00000 => 00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
00001 => 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
00010 => 00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
00011 => 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
00100 => 00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
00101 => 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
00110 => 00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
00111 => 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
01000 => 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
01001 => 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
01010 => 00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
01011 => 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
01100 => 00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
01101 => 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
01110 => 00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
01111 => 10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
10000 => 01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
10001 => 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
10010 => 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
10011 => 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
10100 => 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
10101 => 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
00000000 => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
00000010 => 00000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
00000100 => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
00000110 => 00000011 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
00001000 => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 0
00001010 => 00000101 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
00001100 => 00000110 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
00001110 => 00000111 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
00010000 => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
00010010 => 00001001 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
00010100 => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
00010110 => 00001011 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
00011000 => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
00011010 => 00001101 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
00011100 => 00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
00011110 => 00001111 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
00100000 => 00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
00100010 => 00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
00100100 => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
00100110 => 00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
00101000 => 00010100 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
00101010 => 00010101 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00101100 => 00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
00101110 => 00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
00110000 => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
00110010 => 00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
00110100 => 00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
00110110 => 00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
00111000 => 00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
00111010 => 00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
00111100 => 00011110 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
00111110 => 00011111 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
11000001 => 11100000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
11000011 => 11100001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
11000101 => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
11000111 => 11100011 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
11001001 => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
11001011 => 11100101 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
11001101 => 11100110 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
11001111 => 11100111 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
11010001 => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
11010011 => 11101001 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
11010101 => 11101010 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
11010111 => 11101011 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
11011001 => 11101100 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
11011011 => 11101101 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
11011101 => 11101110 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
11011111 => 11101111 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
Matching statistic: St000069
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00262: Binary words —poset of factors⟶ Posets
St000069: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 20%
St000069: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 20%
Values
0 => ([(0,1)],2)
=> 1 = 0 + 1
1 => ([(0,1)],2)
=> 1 = 0 + 1
00 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> 1 = 0 + 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> 1 = 0 + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> 1 = 0 + 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> 1 = 0 + 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> 1 = 0 + 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> 1 = 0 + 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> 1 = 0 + 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> 1 = 0 + 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> 1 = 0 + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> 1 = 0 + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> 1 = 0 + 1
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> 1 = 0 + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> 1 = 0 + 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> 1 = 0 + 1
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> 1 = 0 + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> 1 = 0 + 1
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> 1 = 0 + 1
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> 1 = 0 + 1
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> 1 = 0 + 1
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> 1 = 0 + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> 1 = 0 + 1
00000010 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0 + 1
00000100 => ([(0,5),(0,6),(1,3),(1,17),(2,14),(3,2),(3,23),(4,1),(4,8),(4,18),(5,21),(5,22),(6,4),(6,21),(6,22),(8,11),(9,13),(10,9),(11,10),(12,7),(13,7),(14,12),(15,9),(15,16),(16,12),(16,13),(17,15),(17,23),(18,17),(18,19),(19,10),(19,15),(20,11),(20,19),(21,18),(21,20),(22,8),(22,20),(23,14),(23,16)],24)
=> ? = 0 + 1
00000101 => ([(0,5),(0,6),(1,4),(1,17),(2,14),(3,2),(3,22),(4,3),(4,21),(5,20),(5,23),(6,1),(6,20),(6,23),(8,11),(9,13),(10,9),(11,10),(12,7),(13,7),(14,12),(15,12),(15,13),(16,9),(16,15),(17,18),(17,21),(18,10),(18,16),(19,11),(19,18),(20,8),(20,19),(21,16),(21,22),(22,14),(22,15),(23,8),(23,17),(23,19)],24)
=> ? = 0 + 1
00000110 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 0 + 1
00000111 => ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
=> ? = 0 + 1
00001000 => ([(0,4),(0,5),(1,13),(2,3),(2,23),(2,24),(3,1),(3,8),(3,19),(4,20),(4,21),(5,2),(5,20),(5,21),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,7),(14,10),(14,17),(15,16),(15,18),(16,9),(16,14),(17,11),(17,12),(18,14),(18,22),(19,13),(19,22),(20,15),(20,24),(21,15),(21,23),(22,7),(22,17),(23,18),(23,19),(24,8),(24,16)],25)
=> ? = 0 + 1
00001010 => ([(0,4),(0,5),(1,3),(1,16),(2,13),(3,2),(3,22),(4,21),(4,24),(5,1),(5,21),(5,24),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,7),(14,10),(14,15),(15,11),(15,12),(16,17),(16,22),(17,14),(17,20),(18,9),(18,14),(19,8),(19,18),(20,7),(20,15),(21,19),(21,23),(22,13),(22,20),(23,8),(23,17),(23,18),(24,16),(24,19),(24,23)],25)
=> ? = 0 + 1
00001011 => ([(0,5),(0,6),(1,15),(2,3),(2,25),(3,4),(3,19),(4,8),(5,1),(5,24),(5,26),(6,2),(6,24),(6,26),(8,11),(9,12),(10,13),(11,14),(12,10),(13,7),(14,7),(15,9),(16,12),(16,22),(17,13),(17,14),(18,22),(18,23),(19,8),(19,23),(20,16),(20,18),(21,9),(21,16),(22,10),(22,17),(23,11),(23,17),(24,20),(24,21),(25,18),(25,19),(26,15),(26,20),(26,21),(26,25)],27)
=> ? = 0 + 1
00001100 => ([(0,5),(0,6),(1,14),(2,1),(2,20),(3,24),(3,26),(4,2),(4,15),(4,25),(5,3),(5,21),(5,22),(6,4),(6,21),(6,22),(8,13),(9,10),(10,11),(11,12),(12,7),(13,7),(14,8),(15,9),(16,11),(16,17),(17,12),(17,13),(18,10),(18,16),(19,16),(19,23),(20,14),(20,23),(21,24),(21,25),(22,15),(22,26),(23,8),(23,17),(24,18),(24,19),(25,19),(25,20),(26,9),(26,18)],27)
=> ? = 0 + 1
00001101 => ([(0,5),(0,6),(1,3),(1,20),(2,15),(3,2),(3,25),(4,21),(4,23),(5,1),(5,24),(5,26),(6,4),(6,24),(6,26),(8,13),(9,12),(10,14),(11,10),(12,11),(13,7),(14,7),(15,8),(16,17),(16,19),(17,10),(17,18),(18,13),(18,14),(19,8),(19,18),(20,16),(20,25),(21,16),(21,22),(22,11),(22,17),(23,12),(23,22),(24,9),(24,23),(25,15),(25,19),(26,9),(26,20),(26,21)],27)
=> ? = 0 + 1
00001110 => ([(0,6),(0,7),(1,3),(1,21),(2,16),(3,2),(3,25),(4,5),(4,17),(4,26),(5,9),(5,18),(6,1),(6,10),(6,19),(7,4),(7,10),(7,19),(9,11),(10,17),(11,12),(12,14),(13,15),(14,8),(15,8),(16,13),(17,9),(18,11),(18,23),(19,21),(19,26),(20,13),(20,24),(21,22),(21,25),(22,20),(22,23),(23,12),(23,24),(24,14),(24,15),(25,16),(25,20),(26,18),(26,22)],27)
=> ? = 0 + 1
00010000 => ([(0,4),(0,5),(1,13),(2,3),(2,23),(2,24),(3,1),(3,8),(3,19),(4,20),(4,21),(5,2),(5,20),(5,21),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,7),(14,10),(14,17),(15,16),(15,18),(16,9),(16,14),(17,11),(17,12),(18,14),(18,22),(19,13),(19,22),(20,15),(20,24),(21,15),(21,23),(22,7),(22,17),(23,18),(23,19),(24,8),(24,16)],25)
=> ? = 0 + 1
00010010 => ([(0,3),(0,4),(1,12),(2,1),(2,18),(2,22),(3,20),(3,21),(4,2),(4,20),(4,21),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,7),(13,14),(13,15),(14,9),(14,16),(15,8),(15,16),(16,10),(16,11),(17,13),(17,23),(18,13),(18,19),(19,6),(19,15),(20,17),(20,18),(21,17),(21,22),(22,12),(22,19),(22,23),(23,6),(23,7),(23,14)],24)
=> ? = 0 + 1
00010011 => ([(0,4),(0,5),(1,15),(2,8),(3,1),(3,16),(3,26),(4,2),(4,24),(4,25),(5,3),(5,24),(5,25),(7,9),(8,7),(9,10),(10,13),(11,14),(12,11),(13,6),(14,6),(15,12),(16,19),(16,23),(17,18),(17,23),(18,12),(18,22),(19,9),(19,21),(20,13),(20,14),(21,10),(21,20),(22,11),(22,20),(23,21),(23,22),(24,16),(24,17),(25,8),(25,17),(25,26),(26,7),(26,15),(26,18),(26,19)],27)
=> ? = 0 + 1
00010100 => ([(0,3),(0,4),(1,14),(2,1),(2,7),(2,15),(3,23),(3,24),(4,2),(4,23),(4,24),(6,8),(7,6),(8,9),(9,12),(10,13),(11,10),(12,5),(13,5),(14,11),(15,14),(15,19),(16,17),(16,18),(17,8),(17,20),(18,20),(18,21),(19,11),(19,21),(20,9),(20,22),(21,10),(21,22),(22,12),(22,13),(23,7),(23,16),(23,25),(24,15),(24,16),(24,25),(25,6),(25,17),(25,18),(25,19)],26)
=> ? = 0 + 1
00010101 => ([(0,3),(0,4),(1,2),(1,20),(2,6),(3,19),(3,23),(4,1),(4,19),(4,23),(6,10),(7,8),(8,11),(9,12),(10,9),(11,5),(12,5),(13,11),(13,12),(14,15),(14,22),(15,7),(15,18),(16,10),(16,17),(17,9),(17,13),(18,8),(18,13),(19,14),(19,21),(20,6),(20,16),(21,15),(21,16),(21,22),(22,7),(22,17),(22,18),(23,14),(23,20),(23,21)],24)
=> ? = 0 + 1
00010110 => ([(0,4),(0,5),(1,15),(2,1),(2,20),(3,8),(3,21),(4,2),(4,26),(4,27),(5,3),(5,26),(5,27),(7,10),(8,11),(9,13),(10,14),(11,12),(12,9),(13,6),(14,6),(15,7),(16,12),(16,24),(17,24),(17,25),(18,7),(18,25),(19,13),(19,14),(20,15),(20,18),(21,11),(21,16),(22,16),(22,17),(23,17),(23,18),(24,9),(24,19),(25,10),(25,19),(26,8),(26,22),(26,23),(27,20),(27,21),(27,22),(27,23)],28)
=> ? = 0 + 1
00011000 => ([(0,4),(0,5),(1,23),(1,24),(2,3),(2,25),(2,26),(3,7),(3,8),(4,1),(4,19),(4,20),(5,2),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,13),(15,14),(16,21),(16,22),(17,11),(17,21),(18,12),(18,22),(19,23),(19,25),(20,24),(20,26),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,16),(24,18),(25,8),(25,17),(26,7),(26,18)],27)
=> ? = 0 + 1
00011001 => ([(0,4),(0,5),(1,14),(2,23),(2,24),(3,1),(3,25),(3,26),(4,2),(4,18),(4,19),(5,3),(5,18),(5,19),(7,10),(8,11),(9,12),(10,13),(11,9),(12,6),(13,6),(14,7),(15,12),(15,13),(16,21),(16,22),(17,11),(17,21),(18,23),(18,25),(19,24),(19,26),(20,7),(20,22),(21,9),(21,15),(22,10),(22,15),(23,16),(23,17),(24,16),(24,20),(25,8),(25,17),(26,8),(26,14),(26,20)],27)
=> ? = 0 + 1
00011010 => ([(0,4),(0,5),(1,15),(2,1),(2,20),(3,21),(3,22),(4,2),(4,26),(4,27),(5,3),(5,26),(5,27),(7,10),(8,11),(9,13),(10,14),(11,12),(12,9),(13,6),(14,6),(15,7),(16,12),(16,24),(17,24),(17,25),(18,7),(18,25),(19,13),(19,14),(20,15),(20,18),(21,17),(21,18),(22,16),(22,17),(23,11),(23,16),(24,9),(24,19),(25,10),(25,19),(26,8),(26,22),(26,23),(27,8),(27,20),(27,21),(27,23)],28)
=> ? = 0 + 1
00011100 => ([(0,5),(0,6),(1,15),(2,4),(2,25),(2,26),(3,1),(3,16),(3,27),(4,19),(4,20),(5,2),(5,21),(5,22),(6,3),(6,21),(6,22),(8,11),(9,12),(10,13),(11,14),(12,10),(13,7),(14,7),(15,8),(16,9),(17,13),(17,14),(18,8),(18,24),(19,23),(19,24),(20,12),(20,23),(21,26),(21,27),(22,16),(22,25),(23,10),(23,17),(24,11),(24,17),(25,9),(25,20),(26,18),(26,19),(27,15),(27,18)],28)
=> ? = 0 + 1
00011110 => ([(0,6),(0,7),(1,2),(1,24),(2,9),(3,5),(3,11),(3,23),(4,18),(4,25),(5,4),(5,17),(5,26),(6,3),(6,12),(6,19),(7,1),(7,12),(7,19),(9,13),(10,16),(11,17),(12,11),(13,14),(14,15),(15,8),(16,8),(17,18),(18,10),(19,23),(19,24),(20,15),(20,16),(21,14),(21,20),(22,13),(22,21),(23,22),(23,26),(24,9),(24,22),(25,10),(25,20),(26,21),(26,25)],27)
=> ? = 0 + 1
00011111 => ([(0,7),(0,8),(1,6),(1,19),(2,4),(2,18),(3,14),(4,15),(5,3),(5,23),(6,5),(6,22),(7,1),(7,20),(8,2),(8,20),(10,11),(11,12),(12,9),(13,9),(14,13),(15,10),(16,12),(16,13),(17,11),(17,16),(18,15),(18,21),(19,21),(19,22),(20,18),(20,19),(21,10),(21,17),(22,17),(22,23),(23,14),(23,16)],24)
=> ? = 0 + 1
00100000 => ([(0,5),(0,6),(1,3),(1,17),(2,14),(3,2),(3,23),(4,1),(4,8),(4,18),(5,21),(5,22),(6,4),(6,21),(6,22),(8,11),(9,13),(10,9),(11,10),(12,7),(13,7),(14,12),(15,9),(15,16),(16,12),(16,13),(17,15),(17,23),(18,17),(18,19),(19,10),(19,15),(20,11),(20,19),(21,18),(21,20),(22,8),(22,20),(23,14),(23,16)],24)
=> ? = 0 + 1
00100010 => ([(0,3),(0,4),(1,2),(1,22),(1,23),(2,16),(2,20),(3,17),(3,18),(4,1),(4,17),(4,18),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,19),(12,21),(13,8),(13,15),(14,9),(14,15),(15,10),(15,11),(16,13),(16,14),(17,12),(17,22),(18,12),(18,23),(19,7),(19,14),(20,6),(20,13),(21,6),(21,7),(22,20),(22,21),(23,16),(23,19)],24)
=> ? = 0 + 1
00100011 => ([(0,4),(0,5),(1,15),(2,3),(2,24),(2,26),(3,16),(3,17),(4,1),(4,18),(4,25),(5,2),(5,18),(5,25),(7,9),(8,10),(9,11),(10,14),(11,13),(12,8),(13,6),(14,6),(15,7),(16,9),(16,21),(17,21),(17,22),(18,20),(18,24),(19,8),(19,22),(20,12),(20,19),(21,11),(21,23),(22,10),(22,23),(23,13),(23,14),(24,17),(24,19),(25,15),(25,20),(25,26),(26,7),(26,12),(26,16)],27)
=> ? = 0 + 1
00100100 => ([(0,2),(0,3),(1,14),(1,15),(2,19),(2,20),(3,1),(3,19),(3,20),(5,8),(6,7),(7,4),(8,4),(9,16),(9,18),(10,17),(10,18),(11,16),(11,17),(12,7),(12,8),(13,9),(13,10),(14,9),(14,11),(15,10),(15,11),(16,6),(16,12),(17,5),(17,12),(18,5),(18,6),(19,13),(19,14),(20,13),(20,15)],21)
=> ? = 0 + 1
00100101 => ([(0,2),(0,3),(1,13),(1,14),(2,22),(2,23),(3,1),(3,22),(3,23),(5,6),(6,7),(7,8),(8,11),(9,12),(10,9),(11,4),(12,4),(13,18),(13,21),(14,15),(14,18),(15,10),(15,19),(16,8),(16,17),(17,11),(17,12),(18,16),(18,19),(19,9),(19,17),(20,6),(20,15),(20,21),(21,7),(21,10),(21,16),(22,5),(22,13),(22,20),(23,5),(23,14),(23,20)],24)
=> ? = 0 + 1
00100110 => ([(0,3),(0,4),(1,22),(1,26),(2,14),(2,23),(3,1),(3,24),(3,25),(4,2),(4,24),(4,25),(6,8),(7,9),(8,10),(9,13),(10,12),(11,7),(12,5),(13,5),(14,6),(15,19),(15,20),(16,8),(16,19),(17,11),(17,18),(18,7),(18,20),(19,10),(19,21),(20,9),(20,21),(21,12),(21,13),(22,15),(22,18),(23,6),(23,16),(24,17),(24,23),(24,26),(25,14),(25,17),(25,22),(26,11),(26,15),(26,16)],27)
=> ? = 0 + 1
00101000 => ([(0,3),(0,4),(1,14),(2,1),(2,7),(2,15),(3,23),(3,24),(4,2),(4,23),(4,24),(6,8),(7,6),(8,9),(9,12),(10,13),(11,10),(12,5),(13,5),(14,11),(15,14),(15,19),(16,17),(16,18),(17,8),(17,20),(18,20),(18,21),(19,11),(19,21),(20,9),(20,22),(21,10),(21,22),(22,12),(22,13),(23,7),(23,16),(23,25),(24,15),(24,16),(24,25),(25,6),(25,17),(25,18),(25,19)],26)
=> ? = 0 + 1
00101010 => ([(0,2),(0,3),(1,11),(2,16),(2,20),(3,1),(3,16),(3,20),(5,9),(6,5),(7,6),(8,10),(9,4),(10,4),(11,7),(12,14),(12,18),(13,9),(13,10),(14,15),(14,19),(15,8),(15,13),(16,12),(16,17),(17,7),(17,14),(17,18),(18,6),(18,15),(18,19),(19,5),(19,8),(19,13),(20,11),(20,12),(20,17)],21)
=> ? = 0 + 1
00101100 => ([(0,3),(0,4),(1,24),(1,25),(2,14),(2,15),(3,1),(3,26),(3,27),(4,2),(4,26),(4,27),(6,8),(7,9),(8,10),(9,12),(10,13),(11,7),(12,5),(13,5),(14,11),(15,6),(16,8),(16,22),(17,11),(17,23),(18,19),(18,23),(19,21),(19,22),(20,12),(20,13),(21,9),(21,20),(22,10),(22,20),(23,7),(23,21),(24,16),(24,19),(25,6),(25,16),(26,14),(26,17),(26,18),(26,24),(27,15),(27,17),(27,18),(27,25)],28)
=> ? = 1 + 1
00101110 => ([(0,4),(0,5),(1,15),(2,3),(2,16),(2,25),(3,7),(3,17),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,9),(8,10),(9,11),(10,13),(11,14),(12,8),(13,6),(14,6),(15,12),(16,7),(17,9),(17,23),(18,12),(18,24),(19,20),(19,24),(20,22),(20,23),(21,13),(21,14),(22,10),(22,21),(23,11),(23,21),(24,8),(24,22),(25,17),(25,20),(26,16),(26,18),(26,19),(27,15),(27,18),(27,19),(27,25)],28)
=> ? = 0 + 1
00101111 => ([(0,5),(0,6),(1,15),(2,3),(2,25),(3,4),(3,19),(4,8),(5,1),(5,24),(5,26),(6,2),(6,24),(6,26),(8,11),(9,12),(10,13),(11,14),(12,10),(13,7),(14,7),(15,9),(16,12),(16,22),(17,13),(17,14),(18,22),(18,23),(19,8),(19,23),(20,16),(20,18),(21,9),(21,16),(22,10),(22,17),(23,11),(23,17),(24,20),(24,21),(25,18),(25,19),(26,15),(26,20),(26,21),(26,25)],27)
=> ? = 0 + 1
00110000 => ([(0,5),(0,6),(1,14),(2,1),(2,20),(3,24),(3,26),(4,2),(4,15),(4,25),(5,3),(5,21),(5,22),(6,4),(6,21),(6,22),(8,13),(9,10),(10,11),(11,12),(12,7),(13,7),(14,8),(15,9),(16,11),(16,17),(17,12),(17,13),(18,10),(18,16),(19,16),(19,23),(20,14),(20,23),(21,24),(21,25),(22,15),(22,26),(23,8),(23,17),(24,18),(24,19),(25,19),(25,20),(26,9),(26,18)],27)
=> ? = 0 + 1
00110010 => ([(0,3),(0,4),(1,21),(1,22),(2,23),(2,26),(3,1),(3,24),(3,25),(4,2),(4,24),(4,25),(6,10),(7,9),(8,6),(9,11),(10,12),(11,13),(12,14),(13,5),(14,5),(15,9),(15,18),(16,10),(16,19),(17,18),(17,19),(18,11),(18,20),(19,12),(19,20),(20,13),(20,14),(21,7),(21,15),(22,15),(22,17),(23,16),(23,17),(24,8),(24,21),(24,26),(25,8),(25,22),(25,23),(26,6),(26,7),(26,16)],27)
=> ? = 0 + 1
00110100 => ([(0,3),(0,4),(1,24),(1,25),(2,14),(2,15),(3,1),(3,26),(3,27),(4,2),(4,26),(4,27),(6,8),(7,9),(8,10),(9,12),(10,13),(11,7),(12,5),(13,5),(14,11),(15,6),(16,8),(16,22),(17,11),(17,23),(18,19),(18,23),(19,21),(19,22),(20,12),(20,13),(21,9),(21,20),(22,10),(22,20),(23,7),(23,21),(24,16),(24,19),(25,6),(25,16),(26,14),(26,17),(26,18),(26,24),(27,15),(27,17),(27,18),(27,25)],28)
=> ? = 1 + 1
00110110 => ([(0,3),(0,4),(1,13),(2,15),(2,22),(3,2),(3,21),(3,24),(4,1),(4,21),(4,24),(6,9),(7,6),(8,11),(9,12),(10,8),(11,5),(12,5),(13,7),(14,17),(14,20),(15,20),(15,23),(16,9),(16,18),(17,10),(17,19),(18,11),(18,12),(19,8),(19,18),(20,16),(20,19),(21,14),(21,15),(22,7),(22,17),(22,23),(23,6),(23,10),(23,16),(24,13),(24,14),(24,22)],25)
=> ? = 0 + 1
00110111 => ([(0,4),(0,5),(1,15),(2,8),(3,1),(3,16),(3,26),(4,2),(4,24),(4,25),(5,3),(5,24),(5,25),(7,9),(8,7),(9,10),(10,13),(11,14),(12,11),(13,6),(14,6),(15,12),(16,19),(16,23),(17,18),(17,23),(18,12),(18,22),(19,9),(19,21),(20,13),(20,14),(21,10),(21,20),(22,11),(22,20),(23,21),(23,22),(24,16),(24,17),(25,8),(25,17),(25,26),(26,7),(26,15),(26,18),(26,19)],27)
=> ? = 0 + 1
00111000 => ([(0,5),(0,6),(1,15),(2,4),(2,25),(2,26),(3,1),(3,16),(3,27),(4,19),(4,20),(5,2),(5,21),(5,22),(6,3),(6,21),(6,22),(8,11),(9,12),(10,13),(11,14),(12,10),(13,7),(14,7),(15,8),(16,9),(17,13),(17,14),(18,8),(18,24),(19,23),(19,24),(20,12),(20,23),(21,26),(21,27),(22,16),(22,25),(23,10),(23,17),(24,11),(24,17),(25,9),(25,20),(26,18),(26,19),(27,15),(27,18)],28)
=> ? = 0 + 1
00111010 => ([(0,4),(0,5),(1,16),(2,3),(2,24),(2,25),(3,17),(3,18),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,9),(9,10),(10,12),(11,13),(12,15),(13,14),(14,6),(15,6),(16,7),(17,21),(17,22),(18,11),(18,21),(19,9),(19,20),(20,10),(20,22),(21,13),(21,23),(22,12),(22,23),(23,14),(23,15),(24,7),(24,18),(25,17),(25,20),(26,8),(26,19),(26,25),(27,8),(27,16),(27,19),(27,24)],28)
=> ? = 0 + 1
00111011 => ([(0,4),(0,5),(1,15),(2,3),(2,24),(2,26),(3,16),(3,17),(4,1),(4,18),(4,25),(5,2),(5,18),(5,25),(7,9),(8,10),(9,11),(10,14),(11,13),(12,8),(13,6),(14,6),(15,7),(16,9),(16,21),(17,21),(17,22),(18,20),(18,24),(19,8),(19,22),(20,12),(20,19),(21,11),(21,23),(22,10),(22,23),(23,13),(23,14),(24,17),(24,19),(25,15),(25,20),(25,26),(26,7),(26,12),(26,16)],27)
=> ? = 0 + 1
00111100 => ([(0,5),(0,6),(1,4),(1,25),(1,26),(2,16),(2,17),(3,22),(3,23),(4,3),(4,18),(4,19),(5,1),(5,20),(5,21),(6,2),(6,20),(6,21),(8,11),(9,10),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,9),(17,8),(18,10),(18,22),(19,11),(19,23),(20,16),(20,25),(21,17),(21,26),(22,12),(22,24),(23,13),(23,24),(24,14),(24,15),(25,9),(25,18),(26,8),(26,19)],27)
=> ? = 0 + 1
00111101 => ([(0,5),(0,6),(1,17),(2,4),(2,20),(2,21),(3,23),(3,24),(4,3),(4,22),(4,25),(5,2),(5,19),(5,26),(6,1),(6,19),(6,26),(8,10),(9,11),(10,15),(11,16),(12,9),(13,8),(14,12),(15,7),(16,7),(17,13),(18,15),(18,16),(19,14),(19,21),(20,13),(20,25),(21,12),(21,22),(22,9),(22,23),(23,11),(23,18),(24,10),(24,18),(25,8),(25,24),(26,14),(26,17),(26,20)],27)
=> ? = 0 + 1
00111110 => ([(0,6),(0,7),(1,16),(2,4),(2,10),(2,20),(3,18),(3,23),(4,5),(4,17),(4,22),(5,3),(5,19),(5,24),(6,2),(6,11),(6,21),(7,1),(7,11),(7,21),(9,12),(10,17),(11,10),(12,13),(13,14),(14,8),(15,8),(16,9),(17,19),(18,15),(19,18),(20,9),(20,22),(21,16),(21,20),(22,12),(22,24),(23,14),(23,15),(24,13),(24,23)],25)
=> ? = 0 + 1
01000000 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0 + 1
01000010 => ([(0,4),(0,5),(1,3),(1,17),(1,18),(2,23),(2,24),(3,2),(3,21),(3,22),(4,19),(4,20),(5,1),(5,19),(5,20),(7,12),(8,11),(9,13),(10,14),(11,9),(12,10),(13,6),(14,6),(15,7),(15,8),(16,13),(16,14),(17,8),(17,21),(18,7),(18,22),(19,15),(19,17),(20,15),(20,18),(21,11),(21,23),(22,12),(22,24),(23,9),(23,16),(24,10),(24,16)],25)
=> ? = 0 + 1
01000011 => ([(0,5),(0,6),(1,17),(2,4),(2,20),(2,21),(3,23),(3,24),(4,3),(4,22),(4,25),(5,2),(5,19),(5,26),(6,1),(6,19),(6,26),(8,10),(9,11),(10,15),(11,16),(12,9),(13,8),(14,12),(15,7),(16,7),(17,13),(18,15),(18,16),(19,14),(19,21),(20,13),(20,25),(21,12),(21,22),(22,9),(22,23),(23,11),(23,18),(24,10),(24,18),(25,8),(25,24),(26,14),(26,17),(26,20)],27)
=> ? = 0 + 1
01000100 => ([(0,3),(0,4),(1,2),(1,22),(1,23),(2,16),(2,20),(3,17),(3,18),(4,1),(4,17),(4,18),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,19),(12,21),(13,8),(13,15),(14,9),(14,15),(15,10),(15,11),(16,13),(16,14),(17,12),(17,22),(18,12),(18,23),(19,7),(19,14),(20,6),(20,13),(21,6),(21,7),(22,20),(22,21),(23,16),(23,19)],24)
=> ? = 0 + 1
01000110 => ([(0,4),(0,5),(1,16),(1,23),(2,3),(2,24),(2,25),(3,18),(3,19),(4,1),(4,26),(4,27),(5,2),(5,26),(5,27),(7,11),(8,10),(9,7),(10,12),(11,13),(12,14),(13,15),(14,6),(15,6),(16,8),(17,10),(17,21),(18,20),(18,21),(19,11),(19,20),(20,13),(20,22),(21,12),(21,22),(22,14),(22,15),(23,8),(23,17),(24,7),(24,19),(25,17),(25,18),(26,9),(26,23),(26,25),(27,9),(27,16),(27,24)],28)
=> ? = 0 + 1
Description
The number of maximal elements of a poset.
Matching statistic: St000052
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
0 => [1] => [1,0]
=> 0
1 => [1] => [1,0]
=> 0
00 => [2] => [1,1,0,0]
=> 0
01 => [1,1] => [1,0,1,0]
=> 0
10 => [1,1] => [1,0,1,0]
=> 0
11 => [2] => [1,1,0,0]
=> 0
000 => [3] => [1,1,1,0,0,0]
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> 0
010 => [1,1,1] => [1,0,1,0,1,0]
=> 0
011 => [1,2] => [1,0,1,1,0,0]
=> 0
100 => [1,2] => [1,0,1,1,0,0]
=> 0
101 => [1,1,1] => [1,0,1,0,1,0]
=> 0
110 => [2,1] => [1,1,0,0,1,0]
=> 0
111 => [3] => [1,1,1,0,0,0]
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
1111 => [4] => [1,1,1,1,0,0,0,0]
=> 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 0
00000101 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
00000110 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
00000111 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
00001001 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
00001011 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 0
00001101 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 0
00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 0
00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 0
00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
00011111 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
00100000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
00100001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
00100010 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 0
00100011 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
00100101 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 0
00100110 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
00100111 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
00101100 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
00101110 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 0
00101111 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
00110001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
00110010 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 0
00110100 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
00110101 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
00110110 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 0
00110111 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
00111000 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
00111001 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0
00111011 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
00111101 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
01000010 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
01000011 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
01000110 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 0
01000111 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000791
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0
1 => [1] => [1,0]
=> [1,0]
=> ? = 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
000 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
011 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
100 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
110 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
111 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0
00001110 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0
00001111 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
00010001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
00010011 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
00010100 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
00010110 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
00010111 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
00011001 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
00011010 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
00011011 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0
00011100 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
00011101 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 0
00100011 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 0
00100100 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
00100101 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
00100110 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0
00101001 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
00101010 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
00101011 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
00101100 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
00101101 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
00110010 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
00110011 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
00110100 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
00110101 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
00110110 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
00111000 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
00111001 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
00111011 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 0
01000100 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
01000110 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
01000111 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0
01001001 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
01001010 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
01001011 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
01001100 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
01001101 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
01010010 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
01010011 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
01010101 => [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]
=> ? = 0
01010110 => [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]
=> ? = 0
01011001 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
01011010 => [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]
=> ? = 0
01011011 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
01011100 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St000218
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Values
0 => [1] => [1,0]
=> [1] => 0
1 => [1] => [1,0]
=> [1] => 0
00 => [2] => [1,1,0,0]
=> [1,2] => 0
01 => [1,1] => [1,0,1,0]
=> [2,1] => 0
10 => [1,1] => [1,0,1,0]
=> [2,1] => 0
11 => [2] => [1,1,0,0]
=> [1,2] => 0
000 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
001 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
010 => [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
011 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
100 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
101 => [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 0
110 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
111 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 0
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 0
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 0
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 0
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0
0000100 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 0
0001000 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => ? = 0
0001001 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ? = 0
0001011 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? = 0
0001111 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 0
0010000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => ? = 0
0010001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,1,2] => ? = 0
0010010 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,1,2] => ? = 0
0010011 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,1,2] => ? = 0
0010100 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,1,2] => ? = 0
0010110 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,1,2] => ? = 0
0010111 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,1,2] => ? = 0
0011000 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,1,2] => ? = 0
0011011 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ? = 0
0011100 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,1,2] => ? = 0
0011101 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => ? = 0
0011110 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,1,2] => ? = 0
0100000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
0100001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 0
0100010 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? = 0
0100011 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 0
0100100 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? = 0
0100101 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 0
0100110 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? = 0
0100111 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 0
0101000 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
0101001 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 0
0101011 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 0
0101100 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? = 0
0101101 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 0
0101110 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 0
0101111 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
0110000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ? = 0
0110001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => ? = 0
0110010 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => ? = 0
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => ? = 0
0110100 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => ? = 0
0110101 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? = 0
0110110 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => ? = 0
0110111 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => ? = 0
0111000 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ? = 0
0111001 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => ? = 0
0111010 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,3,4,1] => ? = 0
0111011 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => ? = 0
0111100 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ? = 0
0111101 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => ? = 0
0111110 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? = 0
1000001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? = 0
1000010 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => ? = 0
1000011 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ? = 0
Description
The number of occurrences of the pattern 213 in a permutation.
The following 448 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000534The number of 2-rises of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000842The breadth of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000920The logarithmic height of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St001394The genus of a permutation. St000647The number of big descents of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000648The number of 2-excedences of a permutation. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000660The number of rises of length at least 3 of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001732The number of peaks visible from the left. St001793The difference between the clique number and the chromatic number of a graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001596The number of two-by-two squares inside a skew partition. St001597The Frobenius rank of a skew partition. St000091The descent variation of a composition. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000657The smallest part of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000805The number of peaks of the associated bargraph. St000816The number of standard composition tableaux of the composition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001272The number of graphs with the same degree sequence. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001577The minimal number of edges to add or remove to make a graph a cograph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000679The pruning number of an ordered tree. St000264The girth of a graph, which is not a tree. St001845The number of join irreducibles minus the rank of a lattice. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St000183The side length of the Durfee square of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000124The cardinality of the preimage of the Simion-Schmidt map. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000666The number of right tethers of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000872The number of very big descents of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000886The number of permutations with the same antidiagonal sums. St000961The shifted major index of a permutation. St000042The number of crossings of a perfect matching. St000296The length of the symmetric border of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000326The position of the first one in a binary word after appending a 1 at the end. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St001301The first Betti number of the order complex associated with the poset. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000640The rank of the largest boolean interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000097The order of the largest clique of the graph. St000355The number of occurrences of the pattern 21-3. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001735The number of permutations with the same set of runs. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001722The number of minimal chains with small intervals between a binary word and the top element. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000403The Szeged index minus the Wiener index of a graph. St000448The number of pairs of vertices of a graph with distance 2. St000516The number of stretching pairs of a permutation. St000552The number of cut vertices of a graph. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000674The number of hills of a Dyck path. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000260The radius of a connected graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000535The rank-width of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000916The packing number of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001271The competition number of a graph. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001829The common independence number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000258The burning number of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001703The villainy of a graph. St000570The Edelman-Greene number of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000918The 2-limited packing number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000219The number of occurrences of the pattern 231 in a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001367The smallest number which does not occur as degree of a vertex in a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000022The number of fixed points of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001260The permanent of an alternating sign matrix. St001570The minimal number of edges to add to make a graph Hamiltonian. St000787The number of flips required to make a perfect matching noncrossing. St001964The interval resolution global dimension of a poset. St000788The number of nesting-similar perfect matchings of a perfect matching. St000929The constant term of the character polynomial of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000160The multiplicity of the smallest part of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000478Another weight of a partition according to Alladi. St000667The greatest common divisor of the parts of the partition. St000143The largest repeated part of a partition. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000068The number of minimal elements in a poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001490The number of connected components of a skew partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call 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 special CNakayama algebra. St001381The fertility of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001487The number of inner corners of a skew partition. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000214The number of adjacencies of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000944The 3-degree of an integer partition. St001571The Cartan determinant of the integer partition. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000488The number of cycles of a permutation of length at most 2. St001061The number of indices that are both descents and recoils of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001850The number of Hecke atoms of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000098The chromatic number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001281The normalized isoperimetric number of a graph. St001309The number of four-cliques in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001638The book thickness of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001316The domatic number of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001546The number of monomials in the Tutte polynomial of a graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St001691The number of kings in a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type.
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