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A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Conditions as in A179254; additionally, if more than 1 part, first difference > first part.

Equivalently, number of partitions for which the sequence of part counts by decreasing part size is 1, 2, 3, ...  - Olivier Gérard, Jul 28 2017

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..10000 (terms 0..200 from Seiichi Manyama)

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

FORMULA

G.f.: Sum_{k>=0} x^(k*(k+1)*(k+2)/6) / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019

EXAMPLE

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)

MATHEMATICA

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 *)

PROG

(Sage)

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

(Ruby)

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

(PARI)

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

CROSSREFS

Cf. A179254 (condition only on differences), A007294 (nondecreasing instead of strictly increasing), A179255, A320382, A320385, A320387, A320388.

Cf. A007862, A240027, A307895, A320509, A320510, A325324, A325357, A325391, A325460.

Sequence in context: A008649 A008650 A062051 * A108711 A261736 A328796

Adjacent sequences:  A179266 A179267 A179268 * A179270 A179271 A179272

KEYWORD

nonn

AUTHOR

Joerg Arndt, Jan 05 2011

STATUS

approved

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Last modified April 22 16:06 EDT 2021. Contains 343177 sequences. (Running on oeis4.)