A179254 -id:A179254 - cp's OEIS frontend

A342517 Number of strict integer partitions of n with strictly increasing first quotients.

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 8, 10, 11, 13, 14, 16, 16, 19, 21, 23, 27, 29, 31, 34, 36, 40, 43, 47, 49, 53, 56, 59, 66, 71, 75, 81, 86, 89, 97, 104, 110, 119, 123, 132, 143, 148, 156, 168, 177, 184, 198, 209, 218, 232, 246, 257, 269, 282, 294

Offset: 0

Gus Wiseman, Mar 20 2021

nonn

Also the number of reversed strict partitions of n with strictly increasing first quotients.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The partition (14,8,5,3,2) has first quotients (4/7,5/8,3/5,2/3) so is not counted under a(32), even though the differences (-6,-3,-2,-1) are strictly increasing.
The a(1) = 1 through a(13) = 10 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
                                   521   81    91    92    A2    A3
                                         621   532   A1    B1    B2
                                               721   632   732   C1
                                                     821   921   643
                                                                 832
                                                                 A21

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for differences instead of quotients is A179254.
The version for chains of divisors is A342086 (non-strict: A057567).
The non-strict ordered version is A342493.
The non-strict version is A342498 (ranking: A342524).
The weakly increasing version is A342516.
The strictly decreasing version is A342518.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A045690 counts sets with maximum n with all adjacent elements y < 2x.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Less@@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

Seiichi Manyama, Oct 12 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.

Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) > p(k) - p(k-1) for all k >= 3.

			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]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..100 from Seiichi Manyama)

Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320387.

Cf. A081489.

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)

A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.

Original entry on oeis.org

2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 6, 6, 2, 2, 1, 1, 3, 10, 6, 5, 2, 2, 1, 1, 4, 11, 11, 6, 4, 2, 2, 1, 1, 2, 16, 13, 10, 5, 4, 2, 2, 1, 1, 6, 17, 19, 12, 9, 4, 4, 2, 2, 1, 1, 2, 24, 24, 18, 11, 8, 4, 4, 2, 2, 1, 1

Offset: 2

Gus Wiseman, Jul 08 2022

The triangle starts with n = 2, and k ranges from 0 to n - 2.

			Triangle begins:
  2
  2  1
  3  1  1
  2  3  1  1
  4  3  2  1  1
  2  6  3  2  1  1
  4  6  6  2  2  1  1
  3 10  6  5  2  2  1  1
  4 11 11  6  4  2  2  1  1
  2 16 13 10  5  4  2  2  1  1
  6 17 19 12  9  4  4  2  2  1  1
  2 24 24 18 11  8  4  4  2  2  1  1
  4 27 34 22 17 10  7  4  4  2  2  1  1
  4 35 39 33 20 15  9  7  4  4  2  2  1  1
  5 39 56 39 30 19 14  8  7  4  4  2  2  1  1
For example, row n = 8 counts the following reversed partitions:
  (8)         (233)      (35)      (125)    (26)    (116)  (17)
  (44)        (1223)     (134)     (11114)  (1115)
  (2222)      (11123)    (224)
  (11111111)  (11222)    (1124)
              (111122)   (1133)
              (1111112)  (111113)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Leading terms are A000005.
Row sums are A000041.
Counts m such that A056239(m) = n and A286470(m) = k.
This is a trimmed version of A238353, which extends to k = n.
For minimum instead of maximum we have A238354.
Ignoring singletons entirely gives A238710.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A115720 and A115994 count partitions by their Durfee square.
A279945 counts partitions by number of distinct differences.
Cf. A064428, A091602, A179254, A238352, A239455, A286469, A325404, A355524, A355526, A355532.

Table[Length[Select[Reverse/@IntegerPartitions[n], If[Length[#]==1,0,Max@@Differences[#]]==k&]],{n,2,15},{k,0,n-2}]

cp's OEIS Frontend

A342517 Number of strict integer partitions of n with strictly increasing first quotients.

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A320388 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing.

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A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.

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Mathematica