searching the database
Your data matches 649 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000247
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> {{1,2}}
=> {{1,2}}
=> 0
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1,3,4},{2}}
=> 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,2,4},{3,6},{5}}
=> 1
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> {{1,4},{2,3,6},{5}}
=> 1
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3,4},{2,6},{5}}
=> 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> {{1,3},{2,6},{4,5}}
=> 0
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1,3},{2,5,6},{4}}
=> 1
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> {{1,5,6},{2,4},{3}}
=> 1
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> {{1,6},{2,4,5},{3}}
=> 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3,5},{4}}
=> 1
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> {{1,2,6},{3,5},{4}}
=> 1
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,2,5},{3},{4,6}}
=> 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> 0
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> {{1,5},{2},{3,4,6}}
=> 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> {{1,4,5},{2},{3,6}}
=> 1
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1,4},{2},{3,5,6}}
=> 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000352
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000461
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4,1,3,2] => 0
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,1,3,2,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,1,3,2,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,1,3,2,5,4] => 0
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,1,4,2,5,3] => 0
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [5,1,4,2,6,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,3,5,4] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,1,6,2,5,4] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,1,6,2,5,3] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,2,5,1,4,3] => 0
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [5,2,6,1,4,3] => 1
Description
The rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Matching statistic: St000481
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2]
=> 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 1
[(1,3),(2,4)]
=> [3,4,1,2] => [2,2]
=> 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4]
=> 0
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,2,2]
=> 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [2,2,2]
=> 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,2]
=> 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [4,2]
=> 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6]
=> 0
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6]
=> 0
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [4,2]
=> 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [2,2,2]
=> 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [2,2,2]
=> 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,2,2]
=> 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [4,2]
=> 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [4,2]
=> 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,2]
=> 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6]
=> 0
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,2]
=> 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000561
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> {{1,2}}
=> {{1,2}}
=> 0
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1,3,4},{2}}
=> 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,2,4},{3,6},{5}}
=> 1
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> {{1,4},{2,3,6},{5}}
=> 1
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3,4},{2,6},{5}}
=> 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> {{1,3},{2,6},{4,5}}
=> 0
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1,3},{2,5,6},{4}}
=> 1
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> {{1,5,6},{2,4},{3}}
=> 1
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> {{1,6},{2,4,5},{3}}
=> 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3,5},{4}}
=> 1
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> {{1,2,6},{3,5},{4}}
=> 1
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,2,5},{3},{4,6}}
=> 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> 0
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> {{1,5},{2},{3,4,6}}
=> 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> {{1,4,5},{2},{3,6}}
=> 1
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1,4},{2},{3,5,6}}
=> 1
Description
The number of occurrences of the pattern {{1,2,3}} in a set partition.
Matching statistic: St000649
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
=> [3,4,1,2] => [4,3,2,1] => 1
[(1,4),(2,3)]
=> [3,4,2,1] => [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,3,2,1,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,2,1,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,6,3,4,1,2] => 0
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,5,3,4,2,1] => 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,5,4,3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [6,5,4,3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,6,3,4,1,2] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [6,5,3,4,2,1] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [6,5,4,3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,5,4,3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St000660
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,1,0,0]
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000864
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
Matching statistic: St000873
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 1
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => 0
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => 0
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 0
Description
The aix statistic of a permutation.
According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then
$$
\operatorname{aix}\pi = \begin{cases}
\operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\
1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\
0 & \text{ if } \beta = \emptyset
\end{cases}\ .
$$
Matching statistic: St001115
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [4,2,1,3] => 1
[(1,3),(2,4)]
=> [3,4,1,2] => [1,3,4,2] => 0
[(1,4),(2,3)]
=> [3,4,2,1] => [3,4,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [6,4,2,1,3,5] => 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [6,1,3,4,2,5] => 1
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [6,3,4,2,1,5] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,2,5,1,6,4] => 0
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [5,3,6,2,4,1] => 0
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [2,4,5,6,3,1] => 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [2,4,5,6,1,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,6,2,4] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,5,1,3,6,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [5,6,2,1,4,3] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [5,3,6,4,1,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,1,5,6,3,2] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [1,4,5,6,3,2] => 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,5,6,3,2,1] => 1
Description
The number of even descents of a permutation.
The following 639 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001801Half the number of preimage-image pairs of different parity in a permutation. St000007The number of saliances of the permutation. St000028The number of stack-sorts needed to sort a permutation. St000183The side length of the Durfee square of an integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000701The protection number of a binary tree. St000783The side length of the largest staircase partition fitting into a partition. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001432The order dimension of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001062The maximal size of a block of a set partition. St000022The number of fixed points of a permutation. St000023The number of inner peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000091The descent variation of a composition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000159The number of distinct parts of the integer partition. St000221The number of strong fixed points of a permutation. St000237The number of small exceedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons 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. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000353The number of inner valleys of a permutation. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000475The number of parts equal to 1 in a partition. St000480The number of lower covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000502The number of successions of a set partitions. St000534The number of 2-rises of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000552The number of cut vertices of a graph. St000562The number of internal points of a set partition. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000648The number of 2-excedences of a permutation. St000650The number of 3-rises of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000664The number of right ropes of a permutation. St000665The number of rafts of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000754The Grundy value for the game of removing nestings in a perfect matching. St000761The number of ascents in an integer composition. St000768The number of peaks in an integer composition. St000779The tier of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000872The number of very big descents of a permutation. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 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. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001153The number of blocks with even minimum in a set partition. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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 $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001394The genus of a permutation. St001403The number of vertical separators in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001423The number of distinct cubes in a binary word. St001424The number of distinct squares in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. 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. St001524The degree of symmetry of a binary word. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000092The number of outer peaks of a permutation. 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. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000253The crossing number of a set partition. St000260The radius of a connected graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000335The difference of lower and upper interactions. St000354The number of recoils of a permutation. St000382The first part of an integer composition. St000397The Strahler number of a rooted tree. St000402Half the size of the symmetry class of a permutation. St000487The length of the shortest cycle of a permutation. St000531The leading coefficient of the rook polynomial of an integer partition. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000669The number of permutations obtained by switching ascents or descents of size 2. St000675The number of centered multitunnels of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000700The protection number of an ordered tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000831The number of indices that are either descents or recoils. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000920The logarithmic height of a Dyck path. St000928The sum of the coefficients of the character polynomial of an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000990The first ascent of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001052The length of the exterior of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001471The magnitude of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001597The Frobenius rank of a skew partition. St001652The length of a longest interval of consecutive numbers. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001732The number of peaks visible from the left. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001792The arboricity of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000147The largest part of an integer partition. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000759The smallest missing part in an integer partition. St000891The number of distinct diagonal sums of a permutation matrix. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001566The length of the longest arithmetic progression in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St000297The number of leading ones in a binary word. St000658The number of rises of length 2 of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St001114The number of odd descents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001948The number of augmented double ascents of a permutation. St000061The number of nodes on the left branch of a binary tree. St000678The number of up steps after the last double rise of a Dyck path. St001330The hat guessing number of a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000331The number of upper interactions of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000782The indicator function of whether a given perfect matching is an L & P matching. St000216The absolute length of a permutation. St000472The sum of the ascent bottoms of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000653The last descent of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. 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. St001346The number of parking functions that give the same permutation. St001716The 1-improper chromatic number of a graph. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000259The diameter of a connected graph. St001638The book thickness of a graph. St000454The largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St000456The monochromatic index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000741The Colin de Verdière graph invariant. 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. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000549The number of odd partial sums of an integer partition. St000628The balance of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. 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. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in 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. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. 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 $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama 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. St001280The number of parts of an integer partition that are at least two. 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. St001387Number of standard Young tableaux of the skew shape tracing the border of the given 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. St001484The number of singletons of an integer partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001657The number of twos in an integer partition. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. 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. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001961The sum of the greatest common divisors of all pairs of parts. 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 $eA$. St000929The constant term of the character polynomial of an integer partition. St000647The number of big descents of a permutation. St000894The trace of an alternating sign matrix. St000455The second largest eigenvalue of a graph if it is integral. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000893The number of distinct diagonal sums of an alternating sign matrix. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001964The interval resolution global dimension of a poset. St001820The size of the image of the pop stack sorting operator. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000284The Plancherel distribution on integer partitions. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000850The number of 1/2-balanced pairs in a poset. St000881The number of short braid edges in the graph of braid moves of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001396Number of triples of incomparable elements in a finite poset. St001480The number of simple summands of the module J^2/J^3. St000527The width of the poset. St000633The size of the automorphism group of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001875The number of simple modules with projective dimension at most 1. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001557The number of inversions of the second entry of a permutation. St001684The reduced word complexity of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000317The cycle descent number of a permutation. St000682The Grundy value of Welter's game on a binary word. St000711The number of big exceedences of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001550The number of inversions between exceedances where the greater exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001565The number of arithmetic progressions of length 2 in a permutation. St001712The number of natural descents of a standard Young tableau. St001769The reflection length of a signed permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001846The number of elements which do not have a complement in the lattice. St001848The atomic length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001871The number of triconnected components of a graph. St001894The depth of a signed permutation. St001903The number of fixed points of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000181The number of connected components of the Hasse diagram for the poset. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000545The number of parabolic double cosets with minimal element being the given permutation. St000635The number of strictly order preserving maps of a poset into itself. St000710The number of big deficiencies of a permutation. St000736The last entry in the first row of a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001427The number of descents of a signed permutation. St001555The order of a signed permutation. St001642The Prague dimension of a graph. St001668The number of points of the poset minus the width of the poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001863The number of weak excedances of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001889The size of the connectivity set of a signed permutation. St001893The flag descent of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001937The size of the center of a parking function. St000075The orbit size of a standard tableau under promotion. St000261The edge connectivity of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000310The minimal degree of a vertex of a graph. St000886The number of permutations with the same antidiagonal sums. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001812The biclique partition number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000822The Hadwiger number of the graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000145The Dyson rank 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000632The jump number of the poset. St000667The greatest common divisor of the parts of the partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. 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. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001301The first Betti number of the order complex associated with the poset. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001541The Gini index of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001763The Hurwitz number of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001902The number of potential covers of a poset. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000307The number of rowmotion orbits of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001720The minimal length of a chain of small intervals in a lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!