- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000207

Mapping

St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 4
[2,1]
 => 3
[1,1,1]
 => 1
[4]
 => 8
[3,1]
 => 7
[2,2]
 => 5
[2,1,1]
 => 4
[1,1,1,1]
 => 1
[5]
 => 16
[4,1]
 => 15
[3,2]
 => 12
[3,1,1]
 => 11
[2,2,1]
 => 7
[2,1,1,1]
 => 5
[1,1,1,1,1]
 => 1
[6]
 => 32
[5,1]
 => 31
[4,2]
 => 26
[4,1,1]
 => 26
[3,3]
 => 23
[3,2,1]
 => 17
[3,1,1,1]
 => 16
[2,2,2]
 => 12
[2,2,1,1]
 => 11
[2,1,1,1,1]
 => 6
[1,1,1,1,1,1]
 => 1
[7]
 => 64
[6,1]
 => 63
[5,2]
 => 54
[5,1,1]
 => 57
[4,3]
 => 45
[4,2,1]
 => 38
[4,1,1,1]
 => 42
[3,3,1]
 => 27
[3,2,2]
 => 33
[3,2,1,1]
 => 30
[3,1,1,1,1]
 => 22
[2,2,2,1]
 => 14
[2,2,1,1,1]
 => 16
[2,1,1,1,1,1]
 => 7
[1,1,1,1,1,1,1]
 => 1
[8]
 => 128
[7,1]
 => 127
[6,2]
 => 110
[6,1,1]
 => 120
[5,3]
 => 89
[5,2,1]
 => 74

Description

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given

$\lambda$ count how many ''integer compositions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has all vertices in integer lattice points.

Matching statistic: St001645
(load all 3 compositions to match this statistic)

Mapping

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 17%

Values

[1]
 => [[1]]
 => [1] => ([],1)
 => 1
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 1
[1,1]
 => [[1],[2]]
 => [1,1] => ([(0,1)],2)
 => 2
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? ∊ {1,4}
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? ∊ {1,4}
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? ∊ {1,5,7,8}
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {1,5,7,8}
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,5,7,8}
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,5,7,8}
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? ∊ {1,7,11,12,15,16}
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,7,11,12,15,16}
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,7,11,12,15,16}
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,7,11,12,15,16}
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,7,11,12,15,16}
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,7,11,12,15,16}
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => ([(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,14,16,22,27,30,33,38,42,45,54,57,63,64}
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [8] => ([],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [1,7] => ([(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [2,6] => ([(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}

Description

The pebbling number of a connected graph.

Matching statistic: St001232
(load all 4 compositions to match this statistic)

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 15%

Values

[1]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? ∊ {1,4}
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {1,4}
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? ∊ {1,5,7,8}
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {1,5,7,8}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ? ∊ {1,5,7,8}
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,5,7,8}
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? ∊ {1,7,11,12,15,16}
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {1,7,11,12,15,16}
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => ? ∊ {1,7,11,12,15,16}
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,7,11,12,15,16}
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => ? ∊ {1,7,11,12,15,16}
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,7,11,12,15,16}
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,11,12,16,17,23,26,26,31,32}
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001632

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 9%●distinct values known / distinct values provided: 2%

Values

[1]
 => [[1]]
 => [1] => ([],1)
 => ? = 1
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 1
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => ? = 2
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ? ∊ {3,4}
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ? ∊ {3,4}
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? ∊ {4,5,7,8}
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {4,5,7,8}
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ? ∊ {4,5,7,8}
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ? ∊ {4,5,7,8}
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ? ∊ {5,7,11,12,15,16}
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ? ∊ {5,7,11,12,15,16}
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ? ∊ {5,7,11,12,15,16}
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ? ∊ {5,7,11,12,15,16}
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ? ∊ {5,7,11,12,15,16}
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => ? ∊ {5,7,11,12,15,16}
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => ? ∊ {6,11,12,16,17,23,26,26,31,32}
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ([(5,6)],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([],7)
 => ? ∊ {7,14,16,22,27,30,33,38,42,45,54,57,63,64}
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
 => ? ∊ {1,8,22,25,26,29,42,42,48,54,54,64,64,73,74,89,94,99,110,120,127,128}

Description

The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.