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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
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-----------------------------------------------------------------------------
Statistic identifier: St000935
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def refines(L,M):
Lpartsums = set( sum(L[:i]) for i in range(1,len(L)) )
Mpartsums = set( sum(M[:i]) for i in range(1,len(M)) )
return Mpartsums.issubset(Lpartsums)
def statistic(M):
return sum(1 for L in Partitions(sum(M)) if refines(L,M) )
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 2
[1,1] => 1
[3] => 3
[2,1] => 2
[1,1,1] => 1
[4] => 5
[3,1] => 3
[2,2] => 3
[2,1,1] => 2
[1,1,1,1] => 1
[5] => 7
[4,1] => 5
[3,2] => 4
[3,1,1] => 3
[2,2,1] => 3
[2,1,1,1] => 2
[1,1,1,1,1] => 1
[6] => 11
[5,1] => 7
[4,2] => 7
[4,1,1] => 5
[3,3] => 5
[3,2,1] => 4
[3,1,1,1] => 3
[2,2,2] => 4
[2,2,1,1] => 3
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 1
[7] => 15
[6,1] => 11
[5,2] => 9
[5,1,1] => 7
[4,3] => 8
[4,2,1] => 7
[4,1,1,1] => 5
[3,3,1] => 5
[3,2,2] => 5
[3,2,1,1] => 4
[3,1,1,1,1] => 3
[2,2,2,1] => 4
[2,2,1,1,1] => 3
[2,1,1,1,1,1] => 2
[1,1,1,1,1,1,1] => 1
[8] => 22
[7,1] => 15
[6,2] => 15
[6,1,1] => 11
[5,3] => 10
[5,2,1] => 9
[5,1,1,1] => 7
[4,4] => 11
[4,3,1] => 8
[4,2,2] => 9
[4,2,1,1] => 7
[4,1,1,1,1] => 5
[3,3,2] => 6
[3,3,1,1] => 5
[3,2,2,1] => 5
[3,2,1,1,1] => 4
[3,1,1,1,1,1] => 3
[2,2,2,2] => 5
[2,2,2,1,1] => 4
[2,2,1,1,1,1] => 3
[2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1] => 1
[9] => 30
[8,1] => 22
[7,2] => 19
[7,1,1] => 15
[6,3] => 17
[6,2,1] => 15
[6,1,1,1] => 11
[5,4] => 13
[5,3,1] => 10
[5,2,2] => 11
[5,2,1,1] => 9
[5,1,1,1,1] => 7
[4,4,1] => 11
[4,3,2] => 9
[4,3,1,1] => 8
[4,2,2,1] => 9
[4,2,1,1,1] => 7
[4,1,1,1,1,1] => 5
[3,3,3] => 7
[3,3,2,1] => 6
[3,3,1,1,1] => 5
[3,2,2,2] => 6
[3,2,2,1,1] => 5
[3,2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => 3
[2,2,2,2,1] => 5
[2,2,2,1,1,1] => 4
[2,2,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1] => 1
[10] => 42
[9,1] => 30
[8,2] => 29
[8,1,1] => 22
[7,3] => 21
[7,2,1] => 19
[7,1,1,1] => 15
[6,4] => 22
[6,3,1] => 17
[6,2,2] => 19
[6,2,1,1] => 15
[6,1,1,1,1] => 11
[5,5] => 15
[5,4,1] => 13
[5,3,2] => 11
[5,3,1,1] => 10
[5,2,2,1] => 11
[5,2,1,1,1] => 9
[5,1,1,1,1,1] => 7
[4,4,2] => 14
[4,4,1,1] => 11
[4,3,3] => 10
[4,3,2,1] => 9
[4,3,1,1,1] => 8
[4,2,2,2] => 11
[4,2,2,1,1] => 9
[4,2,1,1,1,1] => 7
[4,1,1,1,1,1,1] => 5
[3,3,3,1] => 7
[3,3,2,2] => 7
[3,3,2,1,1] => 6
[3,3,1,1,1,1] => 5
[3,2,2,2,1] => 6
[3,2,2,1,1,1] => 5
[3,2,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1] => 3
[2,2,2,2,2] => 6
[2,2,2,2,1,1] => 5
[2,2,2,1,1,1,1] => 4
[2,2,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 56
[10,1] => 42
[9,2] => 38
[9,1,1] => 30
[8,3] => 32
[8,2,1] => 29
[8,1,1,1] => 22
[7,4] => 26
[7,3,1] => 21
[7,2,2] => 23
[7,2,1,1] => 19
[7,1,1,1,1] => 15
[6,5] => 25
[6,4,1] => 22
[6,3,2] => 19
[6,3,1,1] => 17
[6,2,2,1] => 19
[6,2,1,1,1] => 15
[6,1,1,1,1,1] => 11
[5,5,1] => 15
[5,4,2] => 16
[5,4,1,1] => 13
[5,3,3] => 12
[5,3,2,1] => 11
[5,3,1,1,1] => 10
[5,2,2,2] => 13
[5,2,2,1,1] => 11
[5,2,1,1,1,1] => 9
[5,1,1,1,1,1,1] => 7
[4,4,3] => 15
[4,4,2,1] => 14
[4,4,1,1,1] => 11
[4,3,3,1] => 10
[4,3,2,2] => 10
[4,3,2,1,1] => 9
[4,3,1,1,1,1] => 8
[4,2,2,2,1] => 11
[4,2,2,1,1,1] => 9
[4,2,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1] => 5
[3,3,3,2] => 8
[3,3,3,1,1] => 7
[3,3,2,2,1] => 7
[3,3,2,1,1,1] => 6
[3,3,1,1,1,1,1] => 5
[3,2,2,2,2] => 7
[3,2,2,2,1,1] => 6
[3,2,2,1,1,1,1] => 5
[3,2,1,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1,1] => 3
[2,2,2,2,2,1] => 6
[2,2,2,2,1,1,1] => 5
[2,2,2,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 77
[11,1] => 56
[10,2] => 54
[10,1,1] => 42
[9,3] => 42
[9,2,1] => 38
[9,1,1,1] => 30
[8,4] => 41
[8,3,1] => 32
[8,2,2] => 36
[8,2,1,1] => 29
[8,1,1,1,1] => 22
[7,5] => 29
[7,4,1] => 26
[7,3,2] => 23
[7,3,1,1] => 21
[7,2,2,1] => 23
[7,2,1,1,1] => 19
[7,1,1,1,1,1] => 15
[6,6] => 33
[6,5,1] => 25
[6,4,2] => 27
[6,4,1,1] => 22
[6,3,3] => 21
[6,3,2,1] => 19
[6,3,1,1,1] => 17
[6,2,2,2] => 23
[6,2,2,1,1] => 19
[6,2,1,1,1,1] => 15
[6,1,1,1,1,1,1] => 11
[5,5,2] => 17
[5,5,1,1] => 15
[5,4,3] => 17
[5,4,2,1] => 16
[5,4,1,1,1] => 13
[5,3,3,1] => 12
[5,3,2,2] => 12
[5,3,2,1,1] => 11
[5,3,1,1,1,1] => 10
[5,2,2,2,1] => 13
[5,2,2,1,1,1] => 11
[5,2,1,1,1,1,1] => 9
[5,1,1,1,1,1,1,1] => 7
[4,4,4] => 19
[4,4,3,1] => 15
[4,4,2,2] => 17
[4,4,2,1,1] => 14
[4,4,1,1,1,1] => 11
[4,3,3,2] => 11
[4,3,3,1,1] => 10
[4,3,2,2,1] => 10
[4,3,2,1,1,1] => 9
[4,3,1,1,1,1,1] => 8
[4,2,2,2,2] => 13
[4,2,2,2,1,1] => 11
[4,2,2,1,1,1,1] => 9
[4,2,1,1,1,1,1,1] => 7
[4,1,1,1,1,1,1,1,1] => 5
[3,3,3,3] => 9
[3,3,3,2,1] => 8
[3,3,3,1,1,1] => 7
[3,3,2,2,2] => 8
[3,3,2,2,1,1] => 7
[3,3,2,1,1,1,1] => 6
[3,3,1,1,1,1,1,1] => 5
[3,2,2,2,2,1] => 7
[3,2,2,2,1,1,1] => 6
[3,2,2,1,1,1,1,1] => 5
[3,2,1,1,1,1,1,1,1] => 4
[3,1,1,1,1,1,1,1,1,1] => 3
[2,2,2,2,2,2] => 7
[2,2,2,2,2,1,1] => 6
[2,2,2,2,1,1,1,1] => 5
[2,2,2,1,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1,1] => 3
[2,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
[13] => 101
[12,1] => 77
[10,3] => 59
[9,2,2] => 46
[8,5] => 45
[8,4,1] => 41
[8,3,2] => 35
[8,3,1,1] => 32
[7,6] => 37
[7,5,1] => 29
[7,4,2] => 31
[7,3,3] => 25
[6,6,1] => 33
[6,5,2] => 28
[6,5,1,1] => 25
[6,4,3] => 28
[6,4,2,1] => 27
[6,3,2,2] => 21
[6,3,1,1,1,1] => 17
[6,2,2,1,1,1] => 19
[6,1,1,1,1,1,1,1] => 11
[5,5,3] => 18
[5,4,4] => 21
[5,4,3,1] => 17
[5,4,2,2] => 19
[5,4,2,1,1] => 16
[5,4,1,1,1,1] => 13
[5,3,3,2] => 13
[5,3,3,1,1] => 12
[5,3,2,2,1] => 12
[5,3,2,1,1,1] => 11
[5,3,1,1,1,1,1] => 10
[5,2,2,2,1,1] => 13
[5,2,2,1,1,1,1] => 11
[5,2,1,1,1,1,1,1] => 9
[4,4,4,1] => 19
[4,4,3,2] => 16
[4,4,3,1,1] => 15
[4,4,2,2,1] => 17
[4,3,3,3] => 12
[4,3,3,2,1] => 11
[3,3,3,3,1] => 9
[3,3,3,2,2] => 9
[3,3,2,2,2,1] => 8
[3,3,2,1,1,1,1,1] => 6
[3,2,2,2,2,2] => 8
[3,2,2,2,2,1,1] => 7
[3,1,1,1,1,1,1,1,1,1,1] => 3
[2,2,2,2,2,2,1] => 7
[2,2,2,2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[14] => 135
[13,1] => 101
[12,2] => 98
[12,1,1] => 77
[9,5] => 57
[8,6] => 58
[8,5,1] => 45
[8,4,2] => 50
[8,3,1,1,1] => 32
[7,7] => 42
[7,5,2] => 32
[7,4,3] => 32
[6,6,2] => 41
[6,6,1,1] => 33
[6,5,3] => 29
[6,5,1,1,1] => 25
[6,4,4] => 34
[6,4,2,2] => 32
[6,2,2,2,2] => 27
[6,1,1,1,1,1,1,1,1] => 11
[5,5,4] => 21
[5,5,1,1,1,1] => 15
[5,4,3,2] => 18
[5,4,3,1,1] => 17
[5,4,2,2,1] => 19
[5,4,2,1,1,1] => 16
[5,4,1,1,1,1,1] => 13
[5,3,3,3] => 14
[5,3,3,2,1] => 13
[5,3,2,2,2] => 13
[5,3,2,2,1,1] => 12
[5,3,2,1,1,1,1] => 11
[5,3,1,1,1,1,1,1] => 10
[5,2,2,2,2,1] => 15
[5,2,2,1,1,1,1,1] => 11
[4,4,4,2] => 23
[4,4,4,1,1] => 19
[4,4,3,3] => 17
[4,4,3,2,1] => 16
[4,3,2,2,2,1] => 11
[3,3,3,3,2] => 10
[3,3,3,3,1,1] => 9
[3,3,3,2,2,1] => 9
[3,3,2,2,2,2] => 9
[3,3,1,1,1,1,1,1,1,1] => 5
[3,2,2,2,2,1,1,1] => 7
[2,2,2,2,2,2,2] => 8
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[15] => 176
[14,1] => 135
[12,3] => 107
[11,2,2] => 84
[9,6] => 72
[9,5,1] => 57
[8,7] => 65
[8,5,2] => 49
[7,5,3] => 33
[6,6,3] => 44
[6,5,4] => 33
[6,5,1,1,1,1] => 25
[6,4,3,1,1] => 28
[6,3,3,3] => 25
[6,3,1,1,1,1,1,1] => 17
[6,2,2,2,2,1] => 27
[5,5,5] => 23
[5,4,3,3] => 19
[5,4,3,2,1] => 18
[5,4,3,1,1,1] => 17
[5,3,2,2,2,1] => 13
[5,3,2,2,1,1,1] => 12
[5,3,1,1,1,1,1,1,1] => 10
[4,4,4,3] => 24
[4,4,4,1,1,1] => 19
[4,3,3,3,2] => 13
[3,3,3,3,3] => 11
[3,3,3,3,2,1] => 10
[3,3,3,2,2,2] => 10
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[16] => 231
[15,1] => 176
[12,4] => 133
[12,1,1,1,1] => 77
[10,6] => 102
[8,8] => 82
[8,6,2] => 71
[8,5,3] => 50
[7,6,3] => 48
[7,5,3,1] => 33
[6,6,4] => 53
[6,6,2,2] => 49
[6,5,5] => 35
[6,4,3,3] => 30
[5,5,3,3] => 20
[5,5,2,2,2] => 21
[5,4,4,3] => 26
[5,4,3,2,1,1] => 18
[5,4,2,2,2,1] => 22
[4,4,4,4] => 29
[4,4,4,2,2] => 27
[4,4,3,3,2] => 18
[4,3,3,3,3] => 14
[4,3,3,3,2,1] => 13
[3,3,3,3,2,2] => 11
[3,3,3,3,1,1,1,1] => 9
[3,3,2,2,2,2,2] => 10
[2,2,2,2,2,2,2,2] => 9
[2,2,2,2,2,2,1,1,1,1] => 7
[2,2,2,2,1,1,1,1,1,1,1,1] => 5
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
[17] => 297
[9,8] => 99
[8,8,1] => 82
[8,6,3] => 75
[7,5,3,2] => 34
[6,5,3,3] => 31
[6,5,2,2,2] => 34
[6,4,4,3] => 41
[6,4,4,1,1,1] => 34
[6,3,3,3,2] => 27
[6,3,3,3,1,1] => 25
[5,5,4,3] => 25
[5,5,4,1,1,1] => 21
[5,5,2,2,2,1] => 21
[5,4,4,4] => 31
[5,4,3,2,2,1] => 19
[5,3,3,3,2,1] => 15
[4,4,4,3,2] => 25
[4,4,4,3,1,1] => 24
[4,4,4,2,2,1] => 27
[4,4,3,3,3] => 19
[3,3,3,3,3,2] => 12
[3,2,2,2,2,2,2,2] => 10
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 3
[4,4,4,3,2,1] => 25
[5,4,3,3,2,1] => 20
[6,3,3,3,2,1] => 27
[6,5,2,2,2,1] => 34
[5,5,3,3,1,1] => 20
[6,5,4,1,1,1] => 33
[5,5,3,3,2] => 21
[5,5,4,2,2] => 27
[6,4,4,2,2] => 46
[6,5,4,3] => 38
[4,4,4,3,3] => 26
[4,4,4,4,2] => 34
[5,5,4,4] => 29
[6,4,4,4] => 48
[2,2,2,2,2,2,2,2,2] => 10
[3,3,3,3,3,3] => 13
[18] => 385
[6,2,2,2,2,2,2] => 35
[6,6,6] => 70
[4,2,2,2,2,2,2,2] => 19
[10,4,4] => 110
[5,4,4,3,2,1] => 27
[5,5,3,3,2,1] => 21
[5,5,4,2,2,1] => 27
[6,4,4,2,2,1] => 46
[5,5,4,3,1,1] => 25
[6,4,4,3,1,1] => 41
[6,5,3,3,1,1] => 31
[5,5,4,3,2] => 26
[6,4,4,3,2] => 42
[6,5,3,3,2] => 32
[6,5,4,2,2] => 41
[6,5,4,3,1] => 38
[6,5,4,1,1,1,1] => 33
[4,4,4,4,3] => 35
[4,3,3,3,3,3] => 16
[19] => 490
[7,2,2,2,2,2,2] => 39
[7,6,6] => 74
[5,5,4,3,2,1] => 26
[6,4,4,3,2,1] => 42
[6,5,3,3,2,1] => 32
[6,5,4,2,2,1] => 41
[6,5,4,3,1,1] => 38
[6,5,4,3,2] => 39
[6,5,2,2,2,2,1] => 37
[6,5,4,2,1,1,1] => 37
[6,6,2,2,2,2] => 65
[7,5,4,3,1] => 42
[2,2,2,2,2,2,2,2,2,2] => 11
[3,3,3,3,3,3,2] => 14
[4,4,3,3,3,3] => 21
[4,4,4,4,4] => 41
[5,5,5,5] => 31
[20] => 627
[10,10] => 195
[6,5,4,3,2,1] => 39
[6,3,3,3,3,2,1] => 31
[6,5,3,2,2,2,1] => 32
[6,5,4,3,1,1,1] => 38
[7,6,2,2,2,2] => 69
[6,6,5,4] => 67
[3,3,3,3,3,3,3] => 15
[4,4,4,3,3,3] => 28
[21] => 792
[11,7,3] => 153
[4,4,4,4,3,2,1] => 36
[6,4,3,3,3,2,1] => 33
[6,5,4,2,2,2,1] => 45
[6,5,4,3,2,1,1] => 39
[4,4,4,4,3,3] => 37
[5,4,4,4,3,2,1] => 38
[6,5,3,3,3,2,1] => 34
[6,5,4,3,2,2,1] => 40
[6,4,4,4,3,2,1] => 57
[6,5,4,3,3,2,1] => 41
[3,3,3,3,3,3,3,3] => 17
[4,4,4,4,4,4] => 55
[6,6,6,6] => 125
[5,5,5,4,3,2,1] => 34
[6,5,4,4,3,2,1] => 50
[7,6,6,6] => 129
[6,5,5,4,3,2,1] => 46
[6,6,5,4,3,2,1] => 74
[7,6,5,4,3,2] => 78
[3,3,3,3,3,3,3,3,3] => 19
[7,6,5,4,3,2,1] => 78
[6,6,6,6,6] => 201
[4,4,4,4,4,4,4,4] => 89
[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 17
[3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 18
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Created: Aug 11, 2017 at 17:14 by Christian Stump
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Last Updated: Dec 29, 2023 at 18:03 by Martin Rubey