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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000793
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => {{1}}
=> 0
[1,0,1,0]
=> [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 2
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 2
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St001031
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
St001031: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> ? = 0 - 1
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000254
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
St000254: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> {{1}}
=> ? = 0 - 1
[1,0,1,0]
=> {{1},{2}}
=> 0 = 1 - 1
[1,1,0,0]
=> {{1,2}}
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1 = 2 - 1
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000253
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => {{1}}
=> ? = 0 - 1
[1,0,1,0]
=> [2,1] => {{1,2}}
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1 = 2 - 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000444
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,2] => [2]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000251
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [[1]]
=> {{1}}
=> ? = 0 - 1
[1,0,1,0]
=> [1,2] => [[1,2]]
=> {{1,2}}
=> 1 = 2 - 1
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1 = 2 - 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000442
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> ? = 0 - 1
[1,0,1,0]
=> [1,2] => [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000259
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 2
[1,0,1,0,1,0]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {2,2}
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {2,2}
[1,1,0,1,0,0]
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {2,2,2}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2}
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [6,1,3,2,5,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [1,6,3,5,2,4] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [1,6,3,4,2,5] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [3,4,1,5,2,6] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,6] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3] => [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [4,1,5,2,3,6] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [1,7,3,5,2,4,6] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [1,7,3,5,2,6,4] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [1,7,3,5,6,2,4] => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [1,7,3,6,2,4,5] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [1,7,3,4,2,6,5] => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [1,7,3,4,6,2,5] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001200
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 66%●distinct values known / distinct values provided: 40%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 66%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> ? ∊ {1,2}
[1,1,0,0]
=> [2] => [1] => [1,0]
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2}
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> ? ∊ {1,2,2}
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> ? ∊ {1,2,2}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001198
(load all 134 compositions to match this statistic)
(load all 134 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 60%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1] => [] => []
=> ? = 0
[1,0,1,0]
=> [2,1] => [1] => [1,0]
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [1] => [1,0]
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> ? ∊ {1,2,2}
[1,0,1,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> ? ∊ {1,2,2}
[1,1,0,0,1,0]
=> [3,1,2] => [1,2] => [1,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> ? ∊ {1,2,2}
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,2,2,2,3}
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
Description
The 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 $eA$.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St000015The number of peaks of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama 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:
St001530The depth of a Dyck path. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001280The number of parts of an integer partition that are at least two. St001114The number of odd descents of a permutation. St001432The order dimension of the partition. St001545The second Elser number of a connected graph. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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