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Your data matches 74 different statistics following compositions of up to 3 maps.
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Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,2]]
=> 0
[[1],[2]]
=> 1
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 2
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 1
[[1,4],[2],[3]]
=> 2
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 3
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 1
[[1,2,4,5],[3]]
=> 1
[[1,2,3,5],[4]]
=> 1
[[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 1
[[1,3,4],[2,5]]
=> 2
[[1,2,4],[3,5]]
=> 2
[[1,2,3],[4,5]]
=> 1
[[1,4,5],[2],[3]]
=> 2
[[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> 2
[[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> 2
[[1,2,3],[4],[5]]
=> 2
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 2
[[1,2],[3,5],[4]]
=> 2
[[1,3],[2,4],[5]]
=> 3
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 3
[[1,4],[2],[3],[5]]
=> 3
[[1,3],[2],[4],[5]]
=> 3
[[1,2],[3],[4],[5]]
=> 3
[[1],[2],[3],[4],[5]]
=> 4
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 1
[[1,2,4,5,6],[3]]
=> 1
[[1,2,3,5,6],[4]]
=> 1
[[1,2,3,4,6],[5]]
=> 1
[[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000053
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 0
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St001197
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 0
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001506
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 0
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
Description
Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.
Matching statistic: St000010
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1]
=> 1 = 0 + 1
[[1,2]]
=> 0 => [2] => [2]
=> 1 = 0 + 1
[[1],[2]]
=> 1 => [1,1] => [1,1]
=> 2 = 1 + 1
[[1,2,3]]
=> 00 => [3] => [3]
=> 1 = 0 + 1
[[1,3],[2]]
=> 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[[1,2],[3]]
=> 01 => [2,1] => [2,1]
=> 2 = 1 + 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[[1,2,3,4]]
=> 000 => [4] => [4]
=> 1 = 0 + 1
[[1,3,4],[2]]
=> 100 => [1,3] => [3,1]
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [2,2] => [2,2]
=> 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [3,1] => [3,1]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [2,1,1]
=> 3 = 2 + 1
[[1,2],[3,4]]
=> 010 => [2,2] => [2,2]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [2,1,1]
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [2,1,1]
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> 0000 => [5] => [5]
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [4,1]
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [3,2]
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [3,2]
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [4,1]
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [2,2,1]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [3,2]
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [2,2,1]
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [3,2]
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [2,2,1]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [2,2,1]
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [2,2,1]
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [3,1,1]
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [2,2,1]
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [2,2,1]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [2,2,1]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [2,1,1,1]
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [6]
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [5,1]
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [4,2]
=> 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [3,3]
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [4,2]
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [5,1]
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [3,2,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000011
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 1 = 0 + 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 1 = 0 + 1
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000097
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3]]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3]]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3,4]]
=> 000 => [4] => ([],4)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3,4]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> 0000 => [5] => ([],5)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => ([],1)
=> 1 = 0 + 1
[[1,2]]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[[1],[2]]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3]]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[[1,3],[2]]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3]]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[2],[3]]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3,4]]
=> 000 => [4] => ([],4)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3,4]]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> 0000 => [5] => ([],5)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => ([],6)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => ([(4,5)],6)
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => ([(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000288
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => 1 => 1 = 0 + 1
[[1,2]]
=> 0 => [2] => 10 => 1 = 0 + 1
[[1],[2]]
=> 1 => [1,1] => 11 => 2 = 1 + 1
[[1,2,3]]
=> 00 => [3] => 100 => 1 = 0 + 1
[[1,3],[2]]
=> 10 => [1,2] => 110 => 2 = 1 + 1
[[1,2],[3]]
=> 01 => [2,1] => 101 => 2 = 1 + 1
[[1],[2],[3]]
=> 11 => [1,1,1] => 111 => 3 = 2 + 1
[[1,2,3,4]]
=> 000 => [4] => 1000 => 1 = 0 + 1
[[1,3,4],[2]]
=> 100 => [1,3] => 1100 => 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [2,2] => 1010 => 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [3,1] => 1001 => 2 = 1 + 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => 1101 => 3 = 2 + 1
[[1,2],[3,4]]
=> 010 => [2,2] => 1010 => 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => 1110 => 3 = 2 + 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => 1101 => 3 = 2 + 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => 1011 => 3 = 2 + 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => 1111 => 4 = 3 + 1
[[1,2,3,4,5]]
=> 0000 => [5] => 10000 => 1 = 0 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => 11000 => 2 = 1 + 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => 10100 => 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => 10010 => 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => 10001 => 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => 11010 => 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => 10100 => 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => 11001 => 3 = 2 + 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => 10101 => 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => 10010 => 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => 11100 => 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => 11010 => 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => 10110 => 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => 11001 => 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => 10101 => 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => 10011 => 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => 11101 => 4 = 3 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => 11010 => 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => 10110 => 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => 11011 => 4 = 3 + 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => 10101 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => 11110 => 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => 11101 => 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => 11011 => 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => 10111 => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => 11111 => 5 = 4 + 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => 100000 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => 110000 => 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => 101000 => 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => 100100 => 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => 100010 => 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => 100001 => 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => 110100 => 3 = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001068
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> => [1] => [1,0]
=> 1 = 0 + 1
[[1,2]]
=> 0 => [2] => [1,1,0,0]
=> 1 = 0 + 1
[[1],[2]]
=> 1 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[[1,2,3]]
=> 00 => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[[1,3],[2]]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3]]
=> 01 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[1],[2],[3]]
=> 11 => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,3,4]]
=> 000 => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[1,3,4],[2]]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,4],[3]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[4]]
=> 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3,4]]
=> 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2],[4]]
=> 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2],[3],[4]]
=> 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,2,4,5],[3]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,3,5],[4]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3,4],[5]]
=> 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,5],[3,4]]
=> 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,4],[3,5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3],[4,5]]
=> 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3,5],[2],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3,4],[2],[5]]
=> 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,2,4],[3],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,2,3],[4],[5]]
=> 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3,5],[4]]
=> 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4],[2],[3],[5]]
=> 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[[1,2,4,5,6],[3]]
=> 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[1,2,3,5,6],[4]]
=> 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2,3,4,5],[6]]
=> 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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:
St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph 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. St001963The tree-depth of a graph. St001812The biclique partition number of a graph. St000306The bounce count of a Dyck path. St000167The number of leaves of an ordered tree. St000245The number of ascents of a permutation. St000354The number of recoils of a permutation. St000470The number of runs in a permutation. St000662The staircase size of the code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000021The number of descents of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St000015The number of peaks of a Dyck path. St000822The Hadwiger number of the graph. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000159The number of distinct parts of the integer partition. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001727The number of invisible inversions of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001896The number of right descents of a signed permutations.
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