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Your data matches 740 different statistics following compositions of up to 3 maps.
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Matching statistic: St000028
(load all 79 compositions to match this statistic)
(load all 79 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = -1 + 1
[1,0,1,0]
=> [1,2] => 0 = -1 + 1
[1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 0 + 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]
=> [1,2,3,5,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 1 = 0 + 1
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St000651
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = -1 + 1
[1,0,1,0]
=> [2,1] => 0 = -1 + 1
[1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => 0 = -1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0 = -1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => 1 = 0 + 1
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St001011
(load all 191 compositions to match this statistic)
(load all 191 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = -1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = -1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001090
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001090: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = -1 + 1
[1,0,1,0]
=> [1,2] => 0 = -1 + 1
[1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 0 + 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]
=> [1,2,3,5,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 1 = 0 + 1
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St001192
(load all 104 compositions to match this statistic)
(load all 104 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = -1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = -1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001212
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001212: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = -1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = -1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St001294
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = -1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = -1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.
See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.
Matching statistic: St001296
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001296: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001296: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = -1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> 0 = -1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
Description
The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra.
See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
Matching statistic: St001761
(load all 51 compositions to match this statistic)
(load all 51 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = -1 + 1
[1,0,1,0]
=> [1,2] => 0 = -1 + 1
[1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => 0 = -1 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = -1 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 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]
=> [1,2,3,5,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0 = -1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 1 = 0 + 1
Description
The maximal multiplicity of a letter in a reduced word of a permutation.
For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000058
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = -1 + 2
[1,0,1,0]
=> [1,2] => 1 = -1 + 2
[1,1,0,0]
=> [2,1] => 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,2,3] => 1 = -1 + 2
[1,0,1,1,0,0]
=> [1,3,2] => 2 = 0 + 2
[1,1,0,0,1,0]
=> [2,1,3] => 2 = 0 + 2
[1,1,1,0,0,0]
=> [3,2,1] => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = -1 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 0 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = -1 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = -1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 2 = 0 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 2 = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2 = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 2 = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 2 = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 2 = 0 + 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2 = 0 + 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
The following 730 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000298The order dimension or Dushnik-Miller dimension of a poset. St000451The length of the longest pattern of the form k 1 2. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000154The sum of the descent bottoms of a permutation. St000160The multiplicity of the smallest part of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000834The number of right outer peaks of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001092The number of distinct even parts of a partition. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001587Half of the largest even part of an integer partition. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000007The number of saliances of the permutation. St000010The length of the partition. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000470The number of runs in a permutation. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000701The protection number of a binary tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000822The Hadwiger number of the graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001642The Prague dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001963The tree-depth of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000080The rank of the poset. St000081The number of edges of a graph. St000133The "bounce" of a permutation. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000171The degree of the graph. St000183The side length of the Durfee square of an integer partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000290The major index of a binary word. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000340The number of non-final maximal constant sub-paths of length greater than one. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000378The diagonal inversion number of an integer partition. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000537The cutwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000632The jump number of the poset. St000659The number of rises of length at least 2 of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000730The maximal arc length of a set partition. St000741The Colin de Verdière graph invariant. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000769The major index of a composition regarded as a word. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000919The number of maximal left branches of a binary tree. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001427The number of descents of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001479The number of bridges of a graph. St001484The number of singletons of an integer partition. St001485The modular major index of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001649The length of a longest trail in a graph. St001657The number of twos in an integer partition. St001673The degree of asymmetry of an integer composition. St001691The number of kings in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001777The number of weak descents in an integer composition. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St001869The maximum cut size of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001931The weak major index of an integer composition regarded as a word. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000056The decomposition (or block) number 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. St000086The number of subgraphs. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000153The number of adjacent cycles of a permutation. St000166The depth minus 1 of an ordered tree. St000201The number of leaf nodes in a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000239The number of small weak excedances. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000273The domination number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000328The maximum number of child nodes in a tree. St000343The number of spanning subgraphs of a graph. St000345The number of refinements of a partition. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000381The largest part of an integer composition. St000392The length of the longest run 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. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000528The height of a poset. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000676The number of odd rises of a Dyck path. St000679The pruning number of an ordered tree. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000920The logarithmic height of a Dyck path. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St000972The composition number of a graph. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001165Number of simple modules with even projective 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. 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:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. 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. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001471The magnitude of a Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001652The length of a longest interval of consecutive numbers. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001662The length of the longest factor of consecutive numbers in a permutation. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001732The number of peaks visible from the left. St001741The largest integer such that all patterns of this size are contained in the permutation. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001802The number of endomorphisms of a graph. St001814The number of partitions interlacing the given partition. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000094The depth of an ordered tree. St000203The number of external nodes of a binary tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000521The number of distinct subtrees of an ordered tree. St000876The number of factors in the Catalan decomposition of a binary word. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St000485The length of the longest cycle of a permutation. St000251The number of nonsingleton blocks of a set partition. St000354The number of recoils 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. St000640The rank of the largest boolean interval in a poset. St001592The maximal number of simple paths between any two different vertices of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000668The least common multiple of the parts of the partition. St001062The maximal size of a block of a set partition. St000297The number of leading ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000472The sum of the ascent bottoms of a permutation. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000658The number of rises of length 2 of a Dyck path. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000946The sum of the skew hook positions in a Dyck path. St000956The maximal displacement of a permutation. St000989The number of final rises of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000906The length of the shortest maximal chain in a poset. St000933The number of multipartitions of sizes given by an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St001346The number of parking functions that give the same permutation. St001568The smallest positive integer that does not appear twice in the partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001625The Möbius invariant of a lattice. 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 $eA$. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000667The greatest common divisor of the parts of the partition. St001933The largest multiplicity of a part in an integer partition. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000296The length of the symmetric border of a binary word. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000699The toughness times the least common multiple of 1,. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000877The depth of the binary word interpreted as a path. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001248Sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001541The Gini index of an integer partition. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000026The position of the first return of a Dyck path. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000326The position of the first one in a binary word after appending a 1 at the end. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000655The length of the minimal rise of a Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St000847The number of standard Young tableaux whose descent set is the binary word. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of 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. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001281The normalized isoperimetric number of a graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000464The Schultz index of a connected graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St000995The largest even part of an integer partition. St000647The number of big descents of a permutation. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001141The number of occurrences of hills of size 3 in a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000526The number of posets with combinatorially isomorphic order polytopes. St000478Another weight of a partition according to Alladi. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of 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. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000353The number of inner valleys of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000913The number of ways to refine the partition into singletons. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St001469The holeyness of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000732The number of double deficiencies of a permutation. St000908The length of the shortest maximal antichain in a poset. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001890The maximum magnitude of the Möbius function of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001651The Frankl number of a lattice. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St001394The genus of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000779The tier of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001570The minimal number of edges to add to make a graph Hamiltonian. St000023The number of inner peaks of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000624The normalized sum of the minimal distances to a greater element. St000671The maximin edge-connectivity for choosing a subgraph. St000710The number of big deficiencies of a permutation. St000837The number of ascents of distance 2 of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001353The number of prime nodes in the modular decomposition of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000015The number of peaks of a Dyck path. St000099The number of valleys of a permutation, including the boundary. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000619The number of cyclic descents of a permutation. St000700The protection number of an ordered tree. St000907The number of maximal antichains of minimal length in a poset. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001734The lettericity of a graph. St000806The semiperimeter of the associated bargraph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St000145The Dyson rank of a partition. St001569The maximal modular displacement of a permutation. St001948The number of augmented double ascents of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001896The number of right descents of a signed permutations. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001864The number of excedances of a signed permutation. St001597The Frobenius rank of a skew partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001870The number of positive entries followed by a negative entry in a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. 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 $eA$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000905The number of different multiplicities of parts of an integer composition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000307The number of rowmotion orbits of a poset. St000891The number of distinct diagonal sums of a permutation matrix. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001561The value of the elementary symmetric function evaluated at 1. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St000665The number of rafts of a permutation. St001115The number of even descents of a permutation. 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. 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. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000871The number of very big ascents of a permutation. St000317The cycle descent number of a permutation. St000649The number of 3-excedences of a permutation. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000232The number of crossings of a set partition. St000538The number of even inversions of a permutation. St000563The number of overlapping pairs of blocks of a set partition. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000836The number of descents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001470The cyclic holeyness of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001862The number of crossings of a signed permutation. St001871The number of triconnected components of a graph. St000502The number of successions of a set partitions. St000694The number of affine bounded permutations that project to a given permutation. St000886The number of permutations with the same antidiagonal sums. St000942The number of critical left to right maxima of the parking functions. St001151The number of blocks with odd minimum. St001152The number of pairs with even minimum in a perfect matching. St001489The maximum of the number of descents and the number of inverse descents. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001735The number of permutations with the same set of runs. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001863The number of weak excedances of a signed permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001566The length of the longest arithmetic progression in a permutation. St000735The last entry on the main diagonal of a standard tableau. St001638The book thickness of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation.
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