Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001243
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Mapping
St001243: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1
[1,0,1,0]
 => 2
[1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => 4
[1,0,1,1,0,0]
 => 6
[1,1,0,0,1,0]
 => 6
[1,1,0,1,0,0]
 => 9
[1,1,1,0,0,0]
 => 15
[1,0,1,0,1,0,1,0]
 => 10
[1,0,1,0,1,1,0,0]
 => 15
[1,0,1,1,0,0,1,0]
 => 15
[1,0,1,1,0,1,0,0]
 => 22
[1,0,1,1,1,0,0,0]
 => 36
[1,1,0,0,1,0,1,0]
 => 15
[1,1,0,0,1,1,0,0]
 => 23
[1,1,0,1,0,0,1,0]
 => 22
[1,1,0,1,0,1,0,0]
 => 33
[1,1,0,1,1,0,0,0]
 => 53
[1,1,1,0,0,0,1,0]
 => 36
[1,1,1,0,0,1,0,0]
 => 53
[1,1,1,0,1,0,0,0]
 => 87
[1,1,1,1,0,0,0,0]
 => 155
[1,0,1,0,1,0,1,0,1,0]
 => 26
[1,0,1,0,1,0,1,1,0,0]
 => 39
[1,0,1,0,1,1,0,0,1,0]
 => 39
[1,0,1,0,1,1,0,1,0,0]
 => 57
[1,0,1,0,1,1,1,0,0,0]
 => 93
[1,0,1,1,0,0,1,0,1,0]
 => 39
[1,0,1,1,0,0,1,1,0,0]
 => 59
[1,0,1,1,0,1,0,0,1,0]
 => 57
[1,0,1,1,0,1,0,1,0,0]
 => 84
[1,0,1,1,0,1,1,0,0,0]
 => 134
[1,0,1,1,1,0,0,0,1,0]
 => 93
[1,0,1,1,1,0,0,1,0,0]
 => 134
[1,0,1,1,1,0,1,0,0,0]
 => 216
[1,0,1,1,1,1,0,0,0,0]
 => 380
[1,1,0,0,1,0,1,0,1,0]
 => 39
[1,1,0,0,1,0,1,1,0,0]
 => 59
[1,1,0,0,1,1,0,0,1,0]
 => 59
[1,1,0,0,1,1,0,1,0,0]
 => 87
[1,1,0,0,1,1,1,0,0,0]
 => 143
[1,1,0,1,0,0,1,0,1,0]
 => 57
[1,1,0,1,0,0,1,1,0,0]
 => 87
[1,1,0,1,0,1,0,0,1,0]
 => 84
[1,1,0,1,0,1,0,1,0,0]
 => 125
[1,1,0,1,0,1,1,0,0,0]
 => 201
[1,1,0,1,1,0,0,0,1,0]
 => 134
[1,1,0,1,1,0,0,1,0,0]
 => 195
[1,1,0,1,1,0,1,0,0,0]
 => 317
[1,1,0,1,1,1,0,0,0,0]
 => 549
Description
The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. In other words, given a Dyck path, there is an associated (directed) unit interval graph $\Gamma$. Consider the expansion $$G_\Gamma(x;q) = \sum_{\kappa: V(G) \to \mathbb{N}_+} x_\kappa q^{\mathrm{asc}(\kappa)}$$ using the notation by Alexandersson and Panova. The function $G_\Gamma(x;q)$ is a so called unicellular LLT polynomial, and a symmetric function. Consider the Schur expansion $$G_\Gamma(x;q+1) = \sum_{\lambda} c^\Gamma_\lambda(q) s_\lambda(x).$$ By a result by Haiman and Grojnowski, all $c^\Gamma_\lambda(q)$ have non-negative integer coefficients. Consider the sum $$S_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1).$$ This statistic is $S_\Gamma$. It is still an open problem to find a combinatorial description of the above Schur expansion, a first step would be to find a family of combinatorial objects to sum over.
Matching statistic: St000034
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 15%
Values
[1,0]
 => [[1],[2]]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => 1 = 2 - 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [4,3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => [6,2,4,3,1,5] => 3 = 4 - 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => [6,2,5,3,4,1] => 5 = 6 - 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [6,4,3,2,1,5] => 5 = 6 - 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => [6,5,3,2,4,1] => ? = 9 - 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [6,5,4,3,2,1] => ? = 15 - 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => [8,2,6,4,3,5,1,7] => ? = 10 - 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => [8,2,7,4,3,5,6,1] => ? = 15 - 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => [8,2,6,5,4,3,1,7] => ? = 15 - 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => [8,2,7,5,4,3,6,1] => ? = 22 - 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => [8,2,7,6,4,5,3,1] => ? = 36 - 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => [8,6,3,4,2,5,1,7] => ? = 15 - 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => [8,7,3,4,2,5,6,1] => ? = 23 - 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => [8,6,3,5,4,2,1,7] => ? = 22 - 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => [3,5,7,8,1,2,4,6] => [8,7,3,5,4,2,6,1] => ? = 33 - 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => [3,6,7,8,1,2,4,5] => [8,7,3,6,4,5,2,1] => ? = 53 - 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => [8,6,5,4,3,2,1,7] => ? = 36 - 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => [4,5,7,8,1,2,3,6] => [8,7,5,4,3,2,6,1] => ? = 53 - 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => [4,6,7,8,1,2,3,5] => [8,7,6,4,3,5,2,1] => ? = 87 - 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [8,7,6,5,4,3,2,1] => ? = 155 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [2,4,6,8,10,1,3,5,7,9] => [10,2,8,4,6,5,3,7,1,9] => ? = 26 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [2,4,6,9,10,1,3,5,7,8] => [10,2,9,4,6,5,3,7,8,1] => ? = 39 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [2,4,7,8,10,1,3,5,6,9] => [10,2,8,4,7,5,6,3,1,9] => ? = 39 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => [2,4,7,9,10,1,3,5,6,8] => [10,2,9,4,7,5,6,3,8,1] => ? = 57 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [2,4,8,9,10,1,3,5,6,7] => [10,2,9,4,8,5,6,7,3,1] => ? = 93 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [2,5,6,8,10,1,3,4,7,9] => [10,2,8,6,5,4,3,7,1,9] => ? = 39 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [2,5,6,9,10,1,3,4,7,8] => [10,2,9,6,5,4,3,7,8,1] => ? = 59 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [2,5,7,8,10,1,3,4,6,9] => [10,2,8,7,5,4,6,3,1,9] => ? = 57 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [2,5,7,9,10,1,3,4,6,8] => [10,2,9,7,5,4,6,3,8,1] => ? = 84 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => [10,2,9,8,5,4,6,7,3,1] => ? = 134 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [2,6,7,8,10,1,3,4,5,9] => [10,2,8,7,6,5,4,3,1,9] => ? = 93 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [2,6,7,9,10,1,3,4,5,8] => [10,2,9,7,6,5,4,3,8,1] => ? = 134 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [2,6,8,9,10,1,3,4,5,7] => [10,2,9,8,6,5,4,7,3,1] => ? = 216 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => [10,2,9,8,7,5,6,4,3,1] => ? = 380 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [3,4,6,8,10,1,2,5,7,9] => [10,8,3,4,6,5,2,7,1,9] => ? = 39 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [3,4,6,9,10,1,2,5,7,8] => [10,9,3,4,6,5,2,7,8,1] => ? = 59 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [3,4,7,8,10,1,2,5,6,9] => [10,8,3,4,7,5,6,2,1,9] => ? = 59 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [3,4,7,9,10,1,2,5,6,8] => [10,9,3,4,7,5,6,2,8,1] => ? = 87 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => [10,9,3,4,8,5,6,7,2,1] => ? = 143 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [3,5,6,8,10,1,2,4,7,9] => [10,8,3,6,5,4,2,7,1,9] => ? = 57 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [3,5,6,9,10,1,2,4,7,8] => [10,9,3,6,5,4,2,7,8,1] => ? = 87 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [3,5,7,8,10,1,2,4,6,9] => [10,8,3,7,5,4,6,2,1,9] => ? = 84 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [3,5,7,9,10,1,2,4,6,8] => [10,9,3,7,5,4,6,2,8,1] => ? = 125 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [3,5,8,9,10,1,2,4,6,7] => [10,9,3,8,5,4,6,7,2,1] => ? = 201 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => [10,8,3,7,6,5,4,2,1,9] => ? = 134 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [3,6,7,9,10,1,2,4,5,8] => [10,9,3,7,6,5,4,2,8,1] => ? = 195 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [3,6,8,9,10,1,2,4,5,7] => [10,9,3,8,6,5,4,7,2,1] => ? = 317 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [3,7,8,9,10,1,2,4,5,6] => [10,9,3,8,7,5,6,4,2,1] => ? = 549 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => [4,5,6,8,10,1,2,3,7,9] => [10,8,6,4,5,3,2,7,1,9] => ? = 93 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [4,5,6,9,10,1,2,3,7,8] => [10,9,6,4,5,3,2,7,8,1] => ? = 143 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => [4,5,7,8,10,1,2,3,6,9] => [10,8,7,4,5,3,6,2,1,9] => ? = 134 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => [4,5,7,9,10,1,2,3,6,8] => [10,9,7,4,5,3,6,2,8,1] => ? = 201 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [[1,2,3,6,7],[4,5,8,9,10]]
 => [4,5,8,9,10,1,2,3,6,7] => [10,9,8,4,5,3,6,7,2,1] => ? = 317 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => [4,6,7,8,10,1,2,3,5,9] => [10,8,7,4,6,5,3,2,1,9] => ? = 216 - 1
Description
The maximum defect over any reduced expression for a permutation and any subexpression.