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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St001312
(load all 9 compositions to match this statistic)
(load all 9 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 7 compositions to match this statistic)
(load all 7 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]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
[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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862}
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]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,8,28,35,56,56,70,83,98,110,125,125,175,185,196,200,226,235,242,250,275,281,332,365,371,376,407,436,476,502,516,572} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 1
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 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]
=> ? ∊ {1,9,9,36,36,44,44,65,84,84,119,119,126,126,140,140,154,154,204,204,209,209,251,254,254,294,294,301,301,319,319,364,364,375,375,399,399,406,406,429,429,431,431,451,451,491,534,534,546,546,624,624,626,626,630,630,658,658,662,666,666,681,681,704,704,709,709,721,721,726,726,794,801,801,802,802,851,851,854,854,858,901,901,931,931,994,994,998,998,1001,1001,1001,1001,1038,1038,1041,1041,1109,1109,1119,1119,1177,1177,1182,1182,1198,1198,1224,1224,1286,1286,1304,1304,1304,1304,1315,1315,1318,1318,1330,1330,1344,1344,1389,1389,1399,1399,1414,1414,1430,1430,1486,1486,1529,1529,1532,1532,1540,1540,1544,1544,1621,1621,1622,1634,1634,1647,1672,1672,1754,1754,1761,1761,1762,1762,1794,1794,1806,1806,1834,1834,1849,1849,1902,1902,1936,1942,1942,1947,1947,1963,1963,1986,1986,2002,2002,2019,2019,2109,2109,2161,2161,2172,2172,2179,2179,2244,2244,2256,2256,2279,2279,2290,2306,2306,2407,2407,2431,2479,2479,2526,2526,2548,2548,2550,2550,2558,2558,2660,2660,2661,2661,2694,2694,2704,2704,2717,2717,2728,2728,2838,2838,2982,2982,2994,2994,3014,3014,3036,3036,3040,3040,3054,3054,3098,3164,3164,3314,3432,3432,3484,3484,3556,3556,3584,3584,3667,3754,3754,4004,4004,4202,4202,4274,4274,4862} - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000108
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]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[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]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[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]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[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]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? ∊ {34,34,40,40,45,45,48,48,53,62,62,62,72,72,75,75,84,90,90,104,104,107,132}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> ? ∊ {27,27,35,35,37,55,55,65,65,69,75,75,81,81,95,95,105,105,107,107,117,117,126,137,137,145,145,157,157,165,165,171,171,180,180,185,185,207,211,211,219,219,233,241,241,252,252,255,255,281,295,295,297,297,345,345,359,359,429}
[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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