A342528 -id:A342528 - 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}]

A380108 Number of distinct partitions of length n binary strings into maximal constant substrings up to permutation.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 29, 48, 75, 118, 179, 272, 403, 596, 865, 1252, 1786, 2538, 3566, 4990, 6918, 9552, 13086, 17856, 24205, 32684, 43881, 58698, 78125, 103618, 136820, 180064, 236031, 308432, 401585, 521340, 674579, 870446, 1119786, 1436798, 1838405, 2346480, 2987204

Offset: 0

Yaroslav Deryavko, Jan 12 2025

nonn

Equivalently, a(n) is the number of partitions of n into parts of two kinds where the number of parts of each kind differ by at most one.

			For n = 3, the partitions are (000), (111), (00, 1), (0, 11), (0, 0, 1), (0, 1, 1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000712, A114921, A342528.

g:= (n, i, t)-> `if`(t>1+n, 0, `if`(n=0, 1, b(n, i, t))):
b:= proc(n, i, t) option remember; add(add(g(n-i*j,
      min(n-i*j, i-1), abs(t+2*h-j)), h=0..j), j=`if`(i=1, n, 0..n/i))
    end:
a:= n-> g(n$2, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 15 2025

g[n_, i_, t_] := If[t > 1+n, 0, If[n == 0, 1, b[n, i, t]]];
b[n_, i_, t_] := b[n, i, t] = Sum[Sum[g[n-i*j,
   Min[n-i*j, i-1], Abs[t+2*h-j]], {h, 0, j}], {j, If[i == 1, n, 0], n/i}];
a[n_] := g[n, n, 0];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 02 2025, after Alois P. Heinz *)

seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(sum(k=0, n\2, polcoef(p, k, y)*(2*polcoef(p, k+1, y) + polcoef(p, k, y))))} \\ Andrew Howroyd, Jan 12 2025

n = 0
while True:
    m = set()
    for i in range(2**n):
        t = bin(i)[2:]
        t = '0' * (n - len(t)) + t + '2'
        l = []
        s = 0
        for j in range(1, n + 1):
            if t[j] != t[j - 1]:
                l.append(t[s:j])
                s = j
        l.sort()
        l = tuple(l)
        m.add(l)
    print(len(m), end=' ')
    n += 1

G.f.: Sum_{k>=0} ([y^k] P(x,y))*([y^k] (1 + 2*y)*P(x,y)), where P(x,y) = Product_{k>=1} 1/(1 - y*x^k). - Andrew Howroyd, Jan 12 2025

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

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