A342525 -id:A342525 - cp's OEIS frontend

A342499 Number of integer partitions of n with strictly decreasing first quotients.

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 11, 14, 15, 18, 20, 23, 26, 31, 34, 39, 42, 45, 51, 58, 65, 70, 78, 83, 91, 102, 111, 122, 133, 145, 158, 170, 182, 202, 217, 231, 248, 268, 285, 307, 332, 354, 374, 404, 436, 468, 502, 537, 576, 618, 654, 694, 737, 782, 830

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

Also the number of reversed partitions of n with strictly decreasing 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 (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)  (32)   (33)   (43)   (44)   (54)
                   (31)  (41)   (42)   (52)   (53)   (63)
                         (221)  (51)   (61)   (62)   (72)
                                (321)  (331)  (71)   (81)
                                              (332)  (432)
                                              (431)  (441)
                                                     (531)
                                                     (3321)

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 A320470.
The ordered version is A342494.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The strict case is A342518.
The Heinz numbers of these partitions are listed by A342525.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with adjacent x < 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Table[Length[Select[IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342518 Number of strict integer partitions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 11, 12, 13, 17, 18, 21, 24, 28, 30, 34, 37, 41, 47, 52, 56, 63, 68, 72, 83, 89, 99, 108, 117, 128, 139, 149, 163, 179, 189, 203, 217, 233, 250, 272, 289, 305, 329, 355, 381, 410, 438, 471, 505, 540, 571, 607, 645, 683, 726

Offset: 0

Gus Wiseman, Mar 20 2021

nonn

Also the number of reversed strict integer partitions of n with strictly decreasing 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 strict partition (12,10,6,3,1) has first quotients (5/6,3/5,1/2,1/3) so is counted under a(32), even though the differences (-2,-4,-3,-2) are not strictly decreasing.
The a(1) = 1 through a(13) = 12 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
                                    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
                                                                   5431

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 A320388.
The version for chains of divisors is A342086 (non-strict: A057567).
The non-strict ordered version is A342494.
The non-strict version is A342499 (ranking: A342525).
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
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@@#&&Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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 prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

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.

For differences instead of quotients we have A325456 (count: A240027).
For multiplicities (prime signature) instead of quotients we have A334965.
The version counting strict divisor chains is A342086.
These partitions are counted by A342498 (strict: A342517, ordered: A342493).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A342494 Number of compositions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 15, 21, 30, 39, 50, 65, 82, 103, 129, 160, 196, 240, 293, 352, 422, 500, 593, 706, 832, 974, 1138, 1324, 1534, 1783, 2054, 2362, 2712, 3108, 3552, 4051, 4606, 5232, 5935, 6713, 7573, 8536, 9597, 10773, 12085, 13534, 15119, 16874, 18809

Offset: 0

Gus Wiseman, Mar 17 2021

nonn

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 composition (1,2,3,4,2) has first quotients (2,3/2,4/3,1/2) so is counted under a(12).
The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,2,1)  (4,1)    (4,2)
                              (1,2,2)  (5,1)
                              (1,3,1)  (1,2,3)
                              (2,2,1)  (1,3,2)
                                       (1,4,1)
                                       (2,3,1)
                                       (3,2,1)
                                       (1,2,2,1)

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

The weakly decreasing version is A069916.
The version for differences instead of quotients is A325548.
The strictly increasing version is A342493.
The unordered version is A342499, ranked by A342525.
The strict unordered version is A342518.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A274199 counts compositions with all adjacent parts x < 2y.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]

a(21)-a(49) from Alois P. Heinz, Mar 18 2021

A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-down partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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 prime indices of 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}

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 counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A242031.
For differences instead of quotients we have A325361 (count: A320466).
These partitions are counted by A342513 (strict: A342519, ordered: A069916).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],GreaterEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

cp's OEIS Frontend

A342499 Number of integer partitions of n with strictly decreasing first quotients.

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A342518 Number of strict integer partitions of n with strictly decreasing first quotients.

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A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

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A342494 Number of compositions of n with strictly decreasing first quotients.

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A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

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