A069916 -id:A069916 - cp's OEIS frontend

A342513 Number of integer partitions of n with weakly decreasing first quotients.

1, 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 15, 20, 21, 24, 28, 29, 33, 40, 44, 49, 57, 61, 65, 77, 84, 87, 99, 106, 115, 132, 141, 152, 167, 180, 193, 212, 228, 246, 274, 290, 309, 338, 357, 382, 412, 439, 463, 498, 536, 569, 608, 648, 693, 743, 790, 839, 903, 949

Offset: 1

Gus Wiseman, Mar 17 2021

nonn

Also called log-concave-down partitions.
Also the number of reversed integer partitions of n with weakly 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 (9,7,4,2,1) has first quotients (7/9,4/7,1/2,1/2) so is counted under a(23).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (111111)  (2221)     (431)
                                               (1111111)  (2222)
                                                          (11111111)

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 ordered version is A069916.
The version for differences instead of quotients is A320466.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The strict case is A342519.
The Heinz numbers of these partitions are A342526.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
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.
A342094 counts partitions with adjacent parts x <= 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

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

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]&]

A300781 Number of log-concave permutations of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 4, 8, 10, 14, 18, 22, 18, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

Pontus von Brömssen, Mar 12 2018

a(n) = 12 for n >= 10, since for these n the only log-concave permutations of {1,...,n} are (1,2,...,n), (1,2,4,5,...,n,3), (1,2,4,6,...,7,5,3), (1,3,4,...,n,2), (1,3,5,...,6,4,2), (1,n,...,3,2), and the reverses of these.

Pontus Andersson (von Brömssen), Log-concave permutations, Manuscript, 2002.
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A069916, A301454.

G.f.: (6*x^10 +4*x^9 -4*x^8 -4*x^7 -4*x^6 -2*x^5 -4*x^4 -2*x^3 -x^2 -1)/(x-1).

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A342513 Number of integer partitions of n with weakly decreasing first quotients.

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

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A300781 Number of log-concave permutations of {1,...,n}.

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