A007294
Number of partitions of n into nonzero triangular numbers.
Original entry on oeis.org
1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 15, 17, 17, 22, 24, 25, 32, 35, 36, 44, 48, 50, 60, 66, 68, 81, 89, 92, 107, 117, 121, 141, 153, 159, 181, 197, 205, 233, 252, 262, 295, 320, 332, 372, 401, 417, 465, 501, 520, 575, 619, 645, 710, 763
Offset: 0
6 = 3+3 = 3+1+1+1 = 1+1+1+1+1+1 so a(6) = 4.
a(7)=4: Four sequences as above are (7,0,..), (5,1,0,..), (3,2,0,..),(2,1,1,0,..). They correspond to the partitions 1^7, 2 1^5, 2^2 1^3, 3 2 1^2 of seven or in the main description to the partitions 1^7, 3 1^4, 3^2 1, 6 1.
From _Gus Wiseman_, May 03 2019: (Start)
The a(1) = 1 through a(9) = 6 partitions using nonzero triangular numbers are the following. The Heinz numbers of these partitions are given by A325363.
1 11 3 31 311 6 61 611 63
111 1111 11111 33 331 3311 333
3111 31111 311111 6111
111111 1111111 11111111 33111
3111111
111111111
The a(1) = 1 through a(10) = 7 partitions with weakly decreasing multiplicities are the following. Equivalent to Matthew Vandermast's comment, the Heinz numbers of these partitions are given by A025487 (products of primorial numbers).
1 11 21 211 2111 321 3211 32111 32211 4321
111 1111 11111 2211 22111 221111 222111 322111
21111 211111 2111111 321111 2221111
111111 1111111 11111111 2211111 3211111
21111111 22111111
111111111 211111111
1111111111
The a(1) = 1 through a(11) = 7 partitions with weakly increasing differences (where the last part is taken to be zero) are the following. The Heinz numbers of these partitions are given by A325362 (A = 10, B = 11).
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(21) (31) (41) (42) (52) (62) (63) (73) (83)
(51) (61) (71) (72) (82) (92)
(321) (421) (521) (81) (91) (A1)
(531) (631) (731)
(621) (721) (821)
(4321) (5321)
(End)
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
- Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006.
- G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
- N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), p. 191.
- Jorge A. Campos-Gonzalez-Angulo, Raphael F. Ribeiro, and Joel Yuen-Zhou, Generalization of the Tavis-Cummings model for multi-level anharmonic systems, arXiv:2101.09475 [physics.optics], 2021.
- Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #P143.
- Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
- James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
- Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, Journal of Integer Sequences, Vol. 7, 2004.
- Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Cf.
A179255 (condition only on differences),
A179269 (parts strictly increasing instead of nondecreasing). -
Joerg Arndt, Mar 22 2011
-
a007294 = p $ tail a000217_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jun 28 2013
-
b:= proc(n,i) option remember;
if n<0 then 0
elif n=0 then 1
elif i=0 then 0
else b(n, i-1) +b(n-i*(i+1)/2, i)
fi
end:
a:= n-> b(n, floor(sqrt(2*n))):
seq(a(n), n=0..100); # Alois P. Heinz, Mar 22 2011
isNondecrP :=proc(L) slp := DIFF(DIFF(L)) ; min(op(%)) >= 0 ; end proc:
A007294 := proc(n) local a, p; a := 0 ; if n = 0 then return 1 ; end if; for p in combinat[partition](n) do if nops(p) = nops(convert(p, set)) then if isNondecrP(p) then if nops(p) =1 then a := a+1 ; elif op(2, p) >= 2*op(1, p) then a := a+1; end if; end if; end if; end do; a ; end proc:
seq(A007294(n), n=0..30) ; # R. J. Mathar, Jan 07 2011
-
CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
(* also *)
t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i == 0, 0, True, b[n, i-1]+b[n-i*(i+1)/2, i]]; a[n_] := b[n, Floor[Sqrt[2*n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[Append[#,0]]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)
nmax = 58; t = Table[PolygonalNumber[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[t, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)
-
N=66; Vec(1/prod(k=1,N,1-x^(k*(k+1)\2))+O(x^N)) \\ Joerg Arndt, Apr 14 2013
-
from functools import lru_cache
from sympy import divisors
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A007294(n):
@lru_cache(maxsize=None)
def a(n): return is_square((n<<3)+1)
@lru_cache(maxsize=None)
def c(n): return sum(d for d in divisors(n,generator=True) if a(d))
return (c(n)+sum(c(k)*A007294(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024
-
def A007294(n):
has_nondecreasing_diffs = lambda x: min(differences(x, 2)) >= 0
special = lambda x: (x[1]-x[0]) >= x[0]
allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_nondecreasing_diffs(x))
return len([1 for x in Partitions(n, max_slope=-1) if allowed(x[::-1])]) # D. S. McNeil, Jan 06 2011
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)
A240026
Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.
Original entry on oeis.org
1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304, 5846, 6373, 6891, 7578, 8227, 8894, 9727, 10550, 11357, 12405, 13404, 14419
Offset: 0
There are a(10) = 27 such partitions of 10:
01: [ 1 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 1 1 2 ]
03: [ 1 1 1 1 1 1 1 3 ]
04: [ 1 1 1 1 1 1 4 ]
05: [ 1 1 1 1 1 2 3 ]
06: [ 1 1 1 1 1 5 ]
07: [ 1 1 1 1 2 4 ]
08: [ 1 1 1 1 6 ]
09: [ 1 1 1 2 5 ]
10: [ 1 1 1 7 ]
11: [ 1 1 2 6 ]
12: [ 1 1 3 5 ]
13: [ 1 1 8 ]
14: [ 1 2 3 4 ]
15: [ 1 2 7 ]
16: [ 1 3 6 ]
17: [ 1 9 ]
18: [ 2 2 2 2 2 ]
19: [ 2 2 2 4 ]
20: [ 2 2 6 ]
21: [ 2 3 5 ]
22: [ 2 8 ]
23: [ 3 3 4 ]
24: [ 3 7 ]
25: [ 4 6 ]
26: [ 5 5 ]
27: [ 10 ]
Cf.
A240027 (strictly increasing differences).
Cf.
A179255 (distinct parts, nondecreasing),
A179254 (distinct parts, strictly increasing).
-
Table[Length[Select[IntegerPartitions[n],OrderedQ[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.reverse
}
cnt
end
def A240026(n)
(0..n).map{|i| f(i)}
end
p A240026(50) # Seiichi Manyama, Oct 13 2018
A240027
Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.
Original entry on oeis.org
1, 1, 2, 2, 4, 4, 5, 7, 9, 9, 13, 14, 16, 20, 23, 25, 32, 34, 38, 45, 51, 55, 65, 70, 77, 89, 99, 106, 122, 131, 143, 161, 177, 189, 211, 229, 248, 272, 298, 317, 349, 378, 406, 440, 479, 511, 554, 597, 640, 686, 744, 792, 850, 913, 973, 1039, 1122, 1189, 1268, 1358, 1444, 1532, 1646, 1742, 1847, 1975, 2094, 2210, 2366
Offset: 0
There are a(15) = 25 such partitions of 15:
01: [ 1 1 2 4 7 ]
02: [ 1 1 2 11 ]
03: [ 1 1 3 10 ]
04: [ 1 1 4 9 ]
05: [ 1 1 13 ]
06: [ 1 2 4 8 ]
07: [ 1 2 12 ]
08: [ 1 3 11 ]
09: [ 1 4 10 ]
10: [ 1 14 ]
11: [ 2 2 3 8 ]
12: [ 2 2 4 7 ]
13: [ 2 2 11 ]
14: [ 2 3 10 ]
15: [ 2 4 9 ]
16: [ 2 13 ]
17: [ 3 3 9 ]
18: [ 3 4 8 ]
19: [ 3 12 ]
20: [ 4 4 7 ]
21: [ 4 11 ]
22: [ 5 10 ]
23: [ 6 9 ]
24: [ 7 8 ]
25: [ 15 ]
Cf.
A240026 (nondecreasing differences).
Cf.
A179255 (distinct parts, nondecreasing),
A179254 (distinct parts, strictly increasing).
-
Table[Length[Select[IntegerPartitions[n],Less@@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.reverse && ary0.uniq == ary0
}
cnt
end
def A240027(n)
(0..n).map{|i| f(i)}
end
p A240027(50) # Seiichi Manyama, Oct 13 2018
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
A320382
Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 7, 8, 10, 10, 12, 16, 14, 16, 20, 20, 23, 27, 26, 29, 35, 34, 38, 44, 43, 48, 55, 53, 59, 68, 67, 74, 83, 79, 88, 100, 98, 106, 118, 117, 127, 142, 139, 149, 164, 165, 179, 192, 191, 206, 226, 224, 240, 260, 257, 277, 301, 299, 319, 344, 346
Offset: 0
There are a(17) = 16 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]
16: [2, 4, 5, 6]
There are a(18) = 20 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: [1, 6, 11]
10: [8, 10]
11: [1, 7, 10]
12: [2, 6, 10]
13: [1, 8, 9]
14: [2, 7, 9]
15: [3, 6, 9]
16: [3, 7, 8]
17: [4, 6, 8]
18: [5, 6, 7]
19: [1, 4, 6, 7]
20: [3, 4, 5, 6]
-
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
}
cnt
end
def A320382(n)
(0..n).map{|i| f(i)}
end
p A320382(50)
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)
A342516
Number of strict integer partitions of n with weakly increasing first quotients.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 14, 15, 17, 17, 21, 22, 26, 29, 31, 32, 35, 38, 42, 45, 48, 51, 58, 59, 63, 70, 76, 80, 88, 94, 98, 105, 113, 121, 129, 133, 143, 153, 159, 166, 183, 189, 195, 210, 221, 231, 248, 262, 273, 284, 298, 312
Offset: 0
The partition (6,3,2,1) has first quotients (1/2,2/3,1/2) so is not counted under a(12), even though the first differences (-3,-1,-1) are weakly increasing.
The a(1) = 1 through a(13) = 11 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
61 71 72 82 83 93 94
421 521 81 91 92 A2 A3
621 532 A1 B1 B2
721 632 732 C1
821 921 643
832
931
A21
The version for differences instead of quotients is
A179255.
The non-strict ordered version is
A342492.
The strictly increasing version is
A342517.
The weakly decreasing version is
A342519.
A000929 counts partitions with all adjacent parts x >= 2y.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with all adjacent parts x <= 2y (strict:
A342095).
Cf.
A000005,
A003114,
A003242,
A005117,
A057567,
A067824,
A238710,
A253249,
A318991,
A318992,
A342528.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&LessEqual@@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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