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Your data matches 719 different statistics following compositions of up to 3 maps.
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Matching statistic: St000012
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000329
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000394
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000617
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
St000617: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The number of global maxima of a Dyck path.
Matching statistic: St000984
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000984: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The number of boxes below precisely one peak.
Imagine that each peak of the Dyck path, drawn with north and east steps, casts a shadow onto the triangular region between it and the diagonal. This statistic is the number of cells which are in the shade of precisely one peak.
Matching statistic: St001295
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St001295: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
Gives the vector space dimension of the homomorphism space between J^2 and J^2.
Matching statistic: St001480
(load all 75 compositions to match this statistic)
(load all 75 compositions to match this statistic)
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St001508
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
St001508: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
Description
The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.
Matching statistic: St001028
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 5
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001290
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001290: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001290: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 5
Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
The following 709 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000676The number of odd rises of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call 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. St001203We associate to 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 Dyck path as follows:
St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000041The number of nestings of a perfect matching. St000053The number of valleys of the Dyck path. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000155The number of exceedances (also excedences) of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000237The number of small exceedances. St000246The number of non-inversions of a permutation. St000306The bounce count of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000883The number of longest increasing subsequences of a permutation. St000911The number of maximal antichains of maximal size in a poset. St000957The number of Bruhat lower covers of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001397Number of pairs of incomparable elements in a finite poset. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000094The depth of an ordered tree. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000643The size of the largest orbit of antichains under Panyushev complementation. St000756The sum of the positions of the left to right maxima of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000673The number of non-fixed points of a permutation. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000843The decomposition number of a perfect matching. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St000925The number of topologically connected components of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001809The index of the step at the first peak of maximal height in a Dyck path. St000004The major index of a permutation. St000010The length of the partition. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000026The position of the first return of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000042The number of crossings of a perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000080The rank of the poset. St000081The number of edges of a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000297The number of leading ones in a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000496The rcs statistic of a set partition. St000502The number of successions of a set partitions. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000797The stat`` of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000845The maximal number of elements covered by an element in a poset. St000874The position of the last double rise in a Dyck path. St000909The number of maximal chains of maximal size in a poset. St000947The major index east count of a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000990The first ascent of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001077The prefix exchange distance of a permutation. St001115The number of even descents of a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001153The number of blocks with even minimum in a set partition. St001161The major index north count of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001428The number of B-inversions of a signed permutation. St001479The number of bridges of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001498The normalised height of a Nakayama algebra with magnitude 1. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001616The number of neutral elements in a lattice. St001718The number of non-empty open intervals in a poset. St001720The minimal length of a chain of small intervals in a lattice. St001869The maximum cut size of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000338The number of pixed points of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. 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. St001727The number of invisible inversions of a permutation. St000203The number of external nodes of a binary tree. St000230Sum of the minimal elements of the blocks of a set partition. St000507The number of ascents of a standard tableau. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000738The first entry in the last row of a standard tableau. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001343The dimension of the reduced incidence algebra of a poset. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001782The order of rowmotion on the set of order ideals of a poset. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000035The number of left outer peaks of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000144The pyramid weight of the Dyck path. St000146The Andrews-Garvan crank of a partition. St000171The degree of the graph. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000271The chromatic index of a graph. St000273The domination number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000312The number of leaves in a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000420The number of Dyck paths that are weakly above a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000504The cardinality of the first block of a set partition. St000527The width of the poset. St000544The cop number of a graph. St000553The number of blocks of a graph. St000616The inversion index of a permutation. St000636The hull number of a graph. St000668The least common multiple of the parts of the partition. St000692Babson and Steingrímsson's statistic of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000708The product of the parts of an integer partition. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods in a graph. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000839The largest opener of a set partition. St000846The maximal number of elements covering an element of a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001050The number of terminal closers of a set partition. St001062The maximal size of a block of a set partition. St001118The acyclic chromatic index of a graph. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001346The number of parking functions that give the same permutation. St001363The Euler characteristic of a graph according to Knill. St001372The length of a longest cyclic run of ones of a binary word. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001379The number of inversions plus the major index of a permutation. St001389The number of partitions of the same length below the given integer partition. St001458The rank of the adjacency matrix of a graph. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001626The number of maximal proper sublattices of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001814The number of partitions interlacing the given partition. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000008The major index of the composition. St000051The size of the left subtree of a binary tree. St000065The number of entries equal to -1 in an alternating sign matrix. St000078The number of alternating sign matrices whose left key is the permutation. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000204The number of internal nodes of a binary tree. St000247The number of singleton blocks of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000272The treewidth of a graph. St000304The load of a permutation. St000305The inverse major index of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000356The number of occurrences of the pattern 13-2. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000386The number of factors DDU in a Dyck path. St000391The sum of the positions of the ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000441The number of successions of a permutation. St000445The number of rises of length 1 of a Dyck path. St000446The disorder of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000463The number of admissible inversions of a permutation. St000475The number of parts equal to 1 in a partition. St000482The (zero)-forcing number of a graph. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000497The lcb statistic of a set partition. St000499The rcb statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000534The number of 2-rises of a permutation. St000536The pathwidth of a graph. St000572The dimension exponent of a set partition. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. 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. St000632The jump number of the poset. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000646The number of big ascents of a permutation. St000667The greatest common divisor of the parts of the partition. St000691The number of changes of a binary word. St000693The modular (standard) major index of a standard tableau. St000730The maximal arc length of a set partition. St000744The length of the path to the largest entry in a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000796The stat' of a permutation. St000798The makl of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000871The number of very big ascents of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001176The size of a partition minus its first part. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001277The degeneracy of a graph. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001311The cyclomatic number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001317The minimal number of occurrences of the forest-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. St001358The largest degree of a regular subgraph of a graph. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001427The number of descents of a signed permutation. St001437The flex of a binary word. St001484The number of singletons of an integer partition. St001571The Cartan determinant of the integer partition. St001584The area statistic between a Dyck path and its bounce path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001671Haglund's hag of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001759The Rajchgot index of a permutation. St001777The number of weak descents in an integer composition. St001820The size of the image of the pop stack sorting operator. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. 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. St000217The number of occurrences of the pattern 312 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000369The dinv deficit of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000519The largest length of a factor maximising the subword complexity. St000565The major index of a set partition. St000648The number of 2-excedences of a permutation. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000747A variant of the major index of a set partition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension 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. St001091The number of parts in an integer partition whose next smaller part has the same size. St001172The number of 1-rises at odd height of a Dyck path. 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. 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. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001377The major index minus the number of inversions of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001411The number of patterns 321 or 3412 in a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001623The number of doubly irreducible elements of a lattice. St001651The Frankl number of a lattice. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001846The number of elements which do not have a complement in the lattice. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000770The major index of an integer partition when read from bottom to top. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St001432The order dimension of the partition. St000481The number of upper covers of a partition in dominance order. St000461The rix statistic of a permutation. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000787The number of flips required to make a perfect matching noncrossing. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000924The number of topologically connected components of a perfect matching. St001875The number of simple modules with projective dimension at most 1. St001396Number of triples of incomparable elements in a finite poset. St000732The number of double deficiencies of a permutation. St000961The shifted major index of a permutation. St001549The number of restricted non-inversions between exceedances. St000002The number of occurrences of the pattern 123 in a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000840The number of closers smaller than the largest opener in a perfect matching. St000022The number of fixed points of a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000647The number of big descents of a permutation. St001430The number of positive entries in a signed permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000222The number of alignments in the permutation. St000327The number of cover relations in a poset. St000367The number of simsun double descents of a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000242The number of indices that are not cyclical small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000060The greater neighbor of the maximum. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000095The number of triangles of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St000241The number of cyclical small excedances. St000873The aix statistic of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001429The number of negative entries in a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001742The difference of the maximal and the minimal degree in a graph. St001948The number of augmented double ascents of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001964The interval resolution global dimension of a poset. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001866The nesting alignments of a signed permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001894The depth of a signed permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001862The number of crossings of a signed permutation. St001896The number of right descents of a signed permutations. St001435The number of missing boxes in the first row. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. St001488The number of corners of a skew partition. St000907The number of maximal antichains of minimal length in a poset. St001394The genus of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001712The number of natural descents of a standard Young tableau. St001821The sorting index of a signed permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000820The number of compositions obtained by rotating the composition. St000670The reversal length of a permutation. St000942The number of critical left to right maxima of the parking functions. 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 $eA$. St001415The length of the longest palindromic prefix of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000215The number of adjacencies of a permutation, zero appended. St000232The number of crossings of a set partition. St000649The number of 3-excedences of a permutation. St000872The number of very big descents of a permutation. St000884The number of isolated descents of a permutation. St000894The trace of an alternating sign matrix. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001433The flag major index of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001768The number of reduced words of a signed permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001892The flag excedance statistic of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000516The number of stretching pairs of a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001520The number of strict 3-descents. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000454The largest eigenvalue of a graph if it is integral. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000862The number of parts of the shifted shape of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000665The number of rafts of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001645The pebbling number of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001096The size of the overlap set of 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. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St001868The number of alignments of type NE of a signed permutation.
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