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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001312
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(load all 2 compositions to match this statistic)
St001312: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 5
[1,2] => 3
[2,1] => 3
[3] => 1
[1,1,1,1] => 14
[1,1,2] => 9
[1,2,1] => 10
[1,3] => 4
[2,1,1] => 9
[2,2] => 6
[3,1] => 4
[4] => 1
[1,1,1,1,1] => 42
[1,1,1,2] => 28
[1,1,2,1] => 32
[1,1,3] => 14
[1,2,1,1] => 32
[1,2,2] => 22
[1,3,1] => 17
[1,4] => 5
[2,1,1,1] => 28
[2,1,2] => 19
[2,2,1] => 22
[2,3] => 10
[3,1,1] => 14
[3,2] => 10
[4,1] => 5
[5] => 1
[1,1,1,1,1,1] => 132
[1,1,1,1,2] => 90
[1,1,1,2,1] => 104
[1,1,1,3] => 48
[1,1,2,1,1] => 107
[1,1,2,2] => 75
[1,1,3,1] => 62
[1,1,4] => 20
[1,2,1,1,1] => 104
[1,2,1,2] => 72
[1,2,2,1] => 84
[1,2,3] => 40
[1,3,1,1] => 62
[1,3,2] => 45
[1,4,1] => 26
[1,5] => 6
[2,1,1,1,1] => 90
[2,1,1,2] => 62
[2,1,2,1] => 72
Description
Number of parabolic noncrossing partitions indexed by the composition.
Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.
Matching statistic: St000420
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 49%
St000420: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 49%
Values
[1] => [1,0]
=> ? = 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 5
[1,2] => [1,0,1,1,0,0]
=> 3
[2,1] => [1,1,0,0,1,0]
=> 3
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 14
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 9
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10
[1,3] => [1,0,1,1,1,0,0,0]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 9
[2,2] => [1,1,0,0,1,1,0,0]
=> 6
[3,1] => [1,1,1,0,0,0,1,0]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 42
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 28
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 32
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 14
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 32
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 22
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 17
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 28
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 19
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 22
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 10
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 14
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 10
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 132
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 90
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 104
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 48
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 107
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 75
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 62
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 20
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 104
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 72
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 84
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 40
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 62
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 45
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 26
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 90
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 62
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 72
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 34
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 572
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 407
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 476
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 242
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 502
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 365
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 332
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 125
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 516
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 371
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 436
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 226
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 376
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 281
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 196
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 56
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 275
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 200
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 235
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 125
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 250
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 185
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 175
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 70
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 110
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 83
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 98
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 56
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 35
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 28
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4862
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3432
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4004
[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2002
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4202
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3036
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2717
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1001
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4274
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3054
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3584
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1834
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3014
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2244
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1529
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 429
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4274
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 49%
St000419: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 49%
Values
[1] => [1,0]
=> ? = 1 - 1
[1,1] => [1,0,1,0]
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 41 = 42 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 27 = 28 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 31 = 32 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 13 = 14 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 31 = 32 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 21 = 22 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 16 = 17 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 18 = 19 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 21 = 22 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 9 = 10 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 131 = 132 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 89 = 90 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 103 = 104 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 47 = 48 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 106 = 107 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 74 = 75 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 61 = 62 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 19 = 20 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 103 = 104 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 71 = 72 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 83 = 84 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 39 = 40 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 61 = 62 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 44 = 45 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 25 = 26 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 89 = 90 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 61 = 62 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 71 = 72 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 33 = 34 - 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 572 - 1
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 407 - 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 476 - 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 242 - 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 502 - 1
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 365 - 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 332 - 1
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 125 - 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 516 - 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 371 - 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 436 - 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 226 - 1
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 376 - 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 281 - 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 196 - 1
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 56 - 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 275 - 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 200 - 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 235 - 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 125 - 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 250 - 1
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 185 - 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 175 - 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 70 - 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 110 - 1
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 83 - 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 98 - 1
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 56 - 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 35 - 1
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 28 - 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4862 - 1
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3432 - 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4004 - 1
[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2002 - 1
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4202 - 1
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3036 - 1
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2717 - 1
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1001 - 1
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4274 - 1
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3054 - 1
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3584 - 1
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1834 - 1
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3014 - 1
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2244 - 1
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1529 - 1
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 429 - 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4274 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 10%●distinct values known / distinct values provided: 8%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 10%●distinct values known / distinct values provided: 8%
Values
[1] => [1,0]
=> []
=> 1
[1,1] => [1,0,1,0]
=> [1]
=> 2
[2] => [1,1,0,0]
=> []
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 5
[1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 3
[2,1] => [1,1,0,0,1,0]
=> [2]
=> 3
[3] => [1,1,1,0,0,0]
=> []
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 14
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 9
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 10
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 9
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 4
[4] => [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 42
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 28
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 32
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 14
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 32
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 22
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 17
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 28
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 19
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 22
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 14
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 10
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 5
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 132
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 90
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 104
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? = 48
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 107
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 75
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? = 62
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 20
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 104
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 72
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 84
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = 40
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? = 62
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 45
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 26
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 6
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 90
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 62
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 72
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 34
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 75
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 53
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 45
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 15
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 48
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 34
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 40
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 20
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 20
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 15
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 6
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 429
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 297
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 345
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 165
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 359
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 255
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 219
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 75
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 359
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 252
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 295
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 145
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 233
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 171
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 107
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> ? = 27
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 345
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 241
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 281
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 137
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 295
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 211
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? = 185
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 65
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 219
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> ? = 157
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? = 185
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 7
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> 21
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> 21
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 7
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 1
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> 8
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> 8
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> 1
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> 9
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> 9
Description
The number of partitions contained in the given partition.
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