A320387
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 3, 5, 7, 4, 7, 8, 6, 8, 11, 7, 9, 13, 9, 11, 16, 12, 15, 18, 13, 17, 20, 17, 21, 24, 19, 24, 30, 22, 28, 34, 26, 34, 38, 30, 37, 43, 37, 42, 48, 41, 50, 58, 48, 55, 64, 53, 64, 71, 59, 73, 81, 69, 79, 89, 79, 90, 101, 87, 100, 111
Offset: 0
There are a(29) = 15 such partitions of 29:
01: [29]
02: [10, 19]
03: [11, 18]
04: [12, 17]
05: [13, 16]
06: [14, 15]
07: [5, 10, 14]
08: [6, 10, 13]
09: [6, 11, 12]
10: [7, 10, 12]
11: [8, 10, 11]
12: [3, 6, 9, 11]
13: [5, 7, 8, 9]
14: [2, 4, 6, 8, 9]
15: [3, 5, 6, 7, 8]
There are a(30) = 18 such partitions of 30:
01: [30]
02: [10, 20]
03: [11, 19]
04: [12, 18]
05: [13, 17]
06: [14, 16]
07: [5, 10, 15]
08: [6, 10, 14]
09: [6, 11, 13]
10: [7, 10, 13]
11: [7, 11, 12]
12: [8, 10, 12]
13: [3, 6, 9, 12]
14: [9, 10, 11]
15: [4, 7, 9, 10]
16: [2, 4, 6, 8, 10]
17: [6, 7, 8, 9]
18: [4, 5, 6, 7, 8]
A053632 counts compositions by weighted sum.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ots[y_]:=Sum[i*y[[i]],{i,Length[y]}];
Table[Length[Select[Range[2^n],ots[prix[#]]==n&]],{n,10}] (* Gus Wiseman, Jan 17 2023 *)
-
seq(n)={Vec(sum(k=1, (sqrtint(8*n+1)+1)\2, my(t=binomial(k,2)); x^t/prod(j=1, k-1, 1 - x^(t-binomial(j,2)) + O(x^(n-t+1)))))} \\ Andrew Howroyd, Jan 22 2023
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary << 0
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0
}
cnt
end
def A320387(n)
(0..n).map{|i| f(i)}
end
p A320387(50)
A320466
Number of partitions of n such that the successive differences of consecutive parts are nonincreasing.
Original entry on oeis.org
1, 1, 2, 3, 4, 5, 7, 7, 9, 12, 12, 13, 18, 17, 21, 25, 24, 27, 34, 33, 38, 44, 43, 47, 58, 56, 62, 70, 70, 78, 90, 84, 96, 109, 108, 118, 132, 127, 140, 158, 158, 167, 189, 185, 204, 221, 218, 236, 260, 261, 282, 301, 299, 322, 358, 350, 376, 405, 404, 432, 472, 466, 500
Offset: 0
There are a(10) = 12 such partitions of 10:
01: [10]
02: [1, 9]
03: [2, 8]
04: [3, 7]
05: [4, 6]
06: [5, 5]
07: [1, 4, 5]
08: [2, 4, 4]
09: [1, 2, 3, 4]
10: [1, 3, 3, 3]
11: [2, 2, 2, 2, 2]
12: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(11) = 13 such partitions of 11:
01: [11]
02: [1, 10]
03: [2, 9]
04: [3, 8]
05: [4, 7]
06: [5, 6]
07: [1, 4, 6]
08: [1, 5, 5]
09: [2, 4, 5]
10: [3, 4, 4]
11: [2, 3, 3, 3]
12: [1, 2, 2, 2, 2, 2]
13: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
Cf.
A320382 (distinct parts, nonincreasing).
-
Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0
}
cnt
end
def A320466(n)
(0..n).map{|i| f(i)}
end
p A320466(50)
A179254
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 14, 15, 19, 21, 22, 28, 30, 32, 39, 42, 44, 54, 58, 61, 72, 77, 82, 96, 102, 108, 124, 133, 141, 160, 171, 180, 203, 218, 230, 256, 273, 289, 320, 342, 361, 395, 423, 447, 486, 520, 548, 594, 635, 669, 721, 769, 811, 871, 928, 978, 1044, 1114
Offset: 0
There are a(17) = 21 such partitions of 17:
01: [ 1 2 4 10 ]
02: [ 1 2 5 9 ]
03: [ 1 2 14 ]
04: [ 1 3 13 ]
05: [ 1 4 12 ]
06: [ 1 5 11 ]
07: [ 1 16 ]
08: [ 2 3 12 ]
09: [ 2 4 11 ]
10: [ 2 5 10 ]
11: [ 2 15 ]
12: [ 3 4 10 ]
13: [ 3 5 9 ]
14: [ 3 14 ]
15: [ 4 5 8 ]
16: [ 4 13 ]
17: [ 5 12 ]
18: [ 6 11 ]
19: [ 7 10 ]
20: [ 8 9 ]
21: [ 17 ]
- _Joerg Arndt_, Mar 31 2014
Cf.
A240026 (partitions with nondecreasing differences),
A240027 (partitions with strictly increasing differences).
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
}
cnt
end
def A179254(n)
(0..n).map{|i| f(i)}
end
p A179254(50) # Seiichi Manyama, Oct 12 2018
-
def A179254(n):
has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
allowed = lambda x: len(x) < 3 or has_increasing_diffs(x)
return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011
A179269
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 10, 10, 10, 13, 14, 14, 18, 19, 19, 23, 25, 25, 30, 32, 33, 38, 41, 42, 48, 52, 54, 60, 65, 67, 75, 81, 84, 92, 99, 103, 113, 121, 126, 136, 147, 153, 165, 177, 184, 197, 213, 221, 236, 253, 264, 280, 301, 313, 331, 355, 371, 390, 418, 435, 458
Offset: 0
a(10) = 5 as there are 5 such partitions of 10: 1 + 3 + 6 = 1 + 9 = 2 + 8 = 3 + 7 = 10.
a(10) = 5 as there are 5 such partitions of 10: 10, 8 + 1 + 1, 6 + 2 + 2, 4 + 3 + 3, 3 + 2 + 2 + 1 + 1 + 1 (second definition).
From _Gus Wiseman_, May 04 2019: (Start)
The a(3) = 1 through a(13) = 7 partitions whose differences are strictly increasing (with the last part taken to be 0) are the following (A = 10, B = 11, C = 12, D = 13). The Heinz numbers of these partitions are given by A325460.
(3) (4) (5) (6) (7) (8) (9) (A) (B) (C) (D)
(31) (41) (51) (52) (62) (72) (73) (83) (93) (94)
(61) (71) (81) (82) (92) (A2) (A3)
(91) (A1) (B1) (B2)
(631) (731) (831) (C1)
(841)
(931)
The a(3) = 1 through a(11) = 5 partitions whose multiplicities form an initial interval of positive integers are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A307895.
(3) (4) (5) (6) (7) (8) (9) (A) (B)
(211) (311) (411) (322) (422) (522) (433) (533)
(511) (611) (711) (622) (722)
(811) (911)
(322111) (422111)
(End)
-
Table[Length@
Select[IntegerPartitions[n],
And @@ Equal[Range[Length[Split[#]]], Length /@ Split[#]] &], {n,
0, 40}] (* Olivier Gérard, Jul 28 2017 *)
Table[Length[Select[IntegerPartitions[n],Less@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 04 2019 *)
-
R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, L[w-1][i-k*t]))); Mat(L)}
seq(n)={my(M=R(n)); concat([1], vector(n, i, vecsum(M[i,])))} \\ Andrew Howroyd, Aug 27 2019
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary << 0
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
}
cnt
end
def A179269(n)
(0..n).map{|i| f(i)}
end
p A179269(50) # Seiichi Manyama, Oct 12 2018
-
def A179269(n):
has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
special = lambda x: (x[1]-x[0]) > x[0]
allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_increasing_diffs(x))
return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011
A179255
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nondecreasing.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 5, 5, 8, 9, 10, 13, 15, 16, 22, 24, 26, 33, 36, 39, 50, 54, 58, 70, 77, 83, 100, 109, 116, 137, 150, 159, 186, 202, 216, 249, 270, 288, 328, 355, 379, 428, 462, 491, 554, 597, 633, 707, 760, 807, 899, 964, 1020, 1127, 1211, 1282, 1412, 1512, 1596, 1750, 1873, 1976, 2160, 2305, 2434, 2652, 2826, 2978
Offset: 0
There are a(17) = 26 such partitions of 17:
01: [ 1 2 3 4 7 ]
02: [ 1 2 3 11 ]
03: [ 1 2 4 10 ] *
04: [ 1 2 5 9 ] *
05: [ 1 2 14 ] *
06: [ 1 3 5 8 ]
07: [ 1 3 13 ] *
08: [ 1 4 12 ] *
09: [ 1 5 11 ] *
10: [ 1 16 ] *
11: [ 2 3 4 8 ]
12: [ 2 3 5 7 ]
13: [ 2 3 12 ] *
14: [ 2 4 11 ] *
15: [ 2 5 10 ] *
16: [ 2 15 ] *
17: [ 3 4 10 ] *
18: [ 3 5 9 ] *
19: [ 3 14 ] *
20: [ 4 5 8 ] *
21: [ 4 13 ] *
22: [ 5 12 ] *
23: [ 6 11 ] *
24: [ 7 10 ] *
25: [ 8 9 ] *
26: [ 17 ] *
The 21 partitions marked with * have strictly increasing differences, see the example for A179254.
- _Joerg Arndt_, Mar 31 2014
Cf.
A240026 (partitions with nondecreasing differences),
A240027 (partitions with strictly increasing differences),
A320382.
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0.reverse
}
cnt
end
def A179255(n)
(0..n).map{|i| f(i)}
end
p A179255(50) # Seiichi Manyama, Oct 12 2018
-
def A179255(n):
has_nondecreasing_diffs = lambda x: min(differences(x,2)) >= 0
allowed = lambda x: len(x) < 3 or has_nondecreasing_diffs(x)
return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011
A320385
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing, and first difference < first part.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 4, 3, 3, 5, 3, 5, 6, 4, 5, 7, 5, 7, 8, 6, 7, 10, 8, 9, 11, 8, 11, 13, 9, 13, 15, 12, 14, 17, 13, 16, 20, 15, 18, 22, 18, 21, 25, 20, 23, 27, 23, 28, 30, 26, 30, 34, 30, 33, 38, 31, 38, 43, 36, 42, 46, 42, 47, 50, 45, 50, 58, 51, 55
Offset: 0
There are a(29) = 10 such partitions of 29:
01: [29]
02: [10, 19]
03: [11, 18]
04: [12, 17]
05: [13, 16]
06: [14, 15]
07: [6, 10, 13]
08: [6, 11, 12]
09: [7, 10, 12]
10: [8, 10, 11]
There are a(30) = 8 such partitions of 30:
01: [30]
02: [11, 19]
03: [12, 18]
04: [13, 17]
05: [14, 16]
06: [6, 11, 13]
07: [7, 11, 12]
08: [4, 7, 9, 10]
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary << 0
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
}
cnt
end
def A320385(n)
(0..n).map{|i| f(i)}
end
p A320385(50)
A342519
Number of strict integer partitions of n with weakly decreasing first quotients.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 12, 14, 15, 18, 18, 21, 25, 29, 32, 38, 40, 44, 51, 57, 61, 66, 73, 77, 89, 97, 104, 115, 124, 135, 147, 160, 174, 193, 206, 218, 238, 254, 272, 293, 313, 331, 353, 381, 408, 436, 468, 499, 532, 569, 610, 651, 694, 735, 783
Offset: 0
The strict partition (10,7,4,2,1) has first quotients (7/10,4/7,1/2,1/2) so is counted under a(24), even though the first differences (-3,-3,-2,-1) are weakly increasing.
The a(1) = 1 through a(13) = 14 strict partitions (A..D = 10..13):
1 2 3 4 5 6 7 8 9 A B C D
21 31 32 42 43 53 54 64 65 75 76
41 51 52 62 63 73 74 84 85
321 61 71 72 82 83 93 94
421 431 81 91 92 A2 A3
432 541 A1 B1 B2
531 631 542 543 C1
4321 641 642 652
731 651 742
741 751
831 841
5421 931
5431
6421
The non-strict ordered version is
A069916.
The version for differences instead of quotients is
A320382.
The weakly increasing version is
A342516.
The strictly decreasing version is
A342518.
A000005 counts constant partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A057567 counts strict chains of divisors with weakly increasing quotients.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with all adjacent parts x <= 2y (strict:
A342095).
A342528 counts compositions with alternately weakly increasing parts.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
A320388
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 14, 16, 19, 18, 21, 25, 23, 26, 31, 29, 33, 38, 36, 40, 46, 44, 49, 56, 53, 58, 66, 64, 70, 77, 76, 82, 92, 89, 96, 106, 104, 113, 123, 120, 130, 142, 141, 149, 162, 160, 172, 186, 184, 195, 211, 210, 223, 238
Offset: 0
There are a(17) = 15 such partitions of 17:
01: [17]
02: [1, 16]
03: [2, 15]
04: [3, 14]
05: [4, 13]
06: [5, 12]
07: [6, 11]
08: [7, 10]
09: [1, 6, 10]
10: [8, 9]
11: [1, 7, 9]
12: [2, 6, 9]
13: [2, 7, 8]
14: [3, 6, 8]
15: [4, 6, 7]
There are a(18) = 14 such partitions of 18:
01: [18]
02: [1, 17]
03: [2, 16]
04: [3, 15]
05: [4, 14]
06: [5, 13]
07: [6, 12]
08: [7, 11]
09: [8, 10]
10: [1, 7, 10]
11: [1, 8, 9]
12: [2, 7, 9]
13: [3, 7, 8]
14: [1, 4, 6, 7]
-
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
}
cnt
end
def A320388(n)
(0..n).map{|i| f(i)}
end
p A320388(50)
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