A325324 -id:A325324 - cp's OEIS frontend

A325325 Number of integer partitions of n with distinct differences between successive parts.

1, 1, 2, 2, 4, 5, 5, 8, 11, 12, 16, 22, 21, 30, 34, 42, 49, 64, 67, 87, 95, 117, 132, 160, 169, 207, 230, 274, 301, 360, 395, 463, 506, 602, 656, 762, 834, 960, 1042, 1220, 1311, 1505, 1643, 1859, 2000, 2341, 2491, 2827, 3083, 3464, 3747, 4302, 4561, 5154

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The Heinz numbers of these partitions are given by A325368.

			The a(0) = 1 through a(9) = 12 partitions:
  ()  (1)  (2)   (3)   (4)    (5)    (6)    (7)    (8)     (9)
           (11)  (21)  (22)   (32)   (33)   (43)   (44)    (54)
                       (31)   (41)   (42)   (52)   (53)    (63)
                       (211)  (221)  (51)   (61)   (62)    (72)
                              (311)  (411)  (322)  (71)    (81)
                                            (331)  (332)   (441)
                                            (421)  (422)   (522)
                                            (511)  (431)   (621)
                                                   (521)   (711)
                                                   (611)   (4221)
                                                   (4211)  (4311)
                                                           (5211)
For example, (5,2,1,1) has differences (-3,-1,0), which are distinct, so (5,2,1,1) is counted under a(9).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400 (terms 0..123 from Alois P. Heinz)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A007862, A049988, A098859, A240026, A240027, A320348, A320466, A320470, A325324, A325349, A325352, A325368, A325404, A325468.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[#]&]],{n,0,30}]

A007294 Number of partitions of n into nonzero triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 15, 17, 17, 22, 24, 25, 32, 35, 36, 44, 48, 50, 60, 66, 68, 81, 89, 92, 107, 117, 121, 141, 153, 159, 181, 197, 205, 233, 252, 262, 295, 320, 332, 372, 401, 417, 465, 501, 520, 575, 619, 645, 710, 763

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Also number of decreasing integer sequences l(1) >= l(2) >= l(3) >= .. 0 such that sum('i*l(i)','i'=1..infinity)=n.
a(n) is also the number of partitions of n such that #{parts equal to i} >= #{parts equal to j} if i <= j.
Also the number of partitions of n (necessarily into distinct parts) where the part sizes are monotonically decreasing (including the last part, which is the difference between the last part and a "part" of size 0). These partitions are the conjugates of the partitions with number of parts of size i increasing. - Franklin T. Adams-Watters, Apr 08 2008
Also partitions with condition as in A179255, and additionally, if more than one part, first difference >= first part: for example, a(10)=7 as there are 7 such partitions of 10:  1+2+3+4 = 1+2+7 = 1+3+6 = 1+9 = 2+8 = 3+7 = 10. - Joerg Arndt, Mar 22 2011
Number of members of A181818 with a bigomega value of n (cf. A001222). - Matthew Vandermast, May 19 2012

			6 = 3+3 = 3+1+1+1 = 1+1+1+1+1+1 so a(6) = 4.
a(7)=4: Four sequences as above are (7,0,..), (5,1,0,..), (3,2,0,..),(2,1,1,0,..). They correspond to the partitions 1^7, 2 1^5, 2^2 1^3, 3 2 1^2 of seven or in the main description to the partitions 1^7, 3 1^4, 3^2 1, 6 1.
From _Gus Wiseman_, May 03 2019: (Start)
The a(1) = 1 through a(9) = 6 partitions using nonzero triangular numbers are the following. The Heinz numbers of these partitions are given by A325363.
  1   11   3     31     311     6        61        611        63
           111   1111   11111   33       331       3311       333
                                3111     31111     311111     6111
                                111111   1111111   11111111   33111
                                                              3111111
                                                              111111111
The a(1) = 1 through a(10) = 7 partitions with weakly decreasing multiplicities are the following. Equivalent to Matthew Vandermast's comment, the Heinz numbers of these partitions are given by A025487 (products of primorial numbers).
  1  11  21   211   2111   321     3211     32111     32211      4321
         111  1111  11111  2211    22111    221111    222111     322111
                           21111   211111   2111111   321111     2221111
                           111111  1111111  11111111  2211111    3211111
                                                      21111111   22111111
                                                      111111111  211111111
                                                                 1111111111
The a(1) = 1 through a(11) = 7 partitions with weakly increasing differences (where the last part is taken to be zero) are the following. The Heinz numbers of these partitions are given by A325362 (A = 10, B = 11).
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)     (B)
            (21)  (31)  (41)  (42)   (52)   (62)   (63)   (73)    (83)
                              (51)   (61)   (71)   (72)   (82)    (92)
                              (321)  (421)  (521)  (81)   (91)    (A1)
                                                   (531)  (631)   (731)
                                                   (621)  (721)   (821)
                                                          (4321)  (5321)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006.
G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), p. 191.
Jorge A. Campos-Gonzalez-Angulo, Raphael F. Ribeiro, and Joel Yuen-Zhou, Generalization of the Tavis-Cummings model for multi-level anharmonic systems, arXiv:2101.09475 [physics.optics], 2021.
Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #P143.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, Journal of Integer Sequences, Vol. 7, 2004.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000217, A051533, A000294.
Cf. A102462.
Row sums of array A176723 and triangle A176724. - Wolfdieter Lang, Jul 19 2010
Cf. A179255 (condition only on differences), A179269 (parts strictly increasing instead of nondecreasing). - Joerg Arndt, Mar 22 2011
Cf. A024940, A280366.
Row sums of A319797.
Cf. A007862, A025487, A240026, A320509, A325324, A325354, A325356, A325362, A325363.

a007294 = p $ tail a000217_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jun 28 2013

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-i*(i+1)/2, i)
      fi
    end:
a:= n-> b(n, floor(sqrt(2*n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 22 2011
isNondecrP :=proc(L) slp := DIFF(DIFF(L)) ; min(op(%)) >= 0 ; end proc:
A007294 := proc(n) local a, p; a := 0 ; if n = 0 then return 1 ; end if; for p in combinat[partition](n) do if nops(p) = nops(convert(p, set)) then if isNondecrP(p) then if nops(p) =1 then a := a+1 ; elif op(2, p) >= 2*op(1, p) then a := a+1; end if; end if; end if; end do; a ; end proc:
seq(A007294(n), n=0..30) ; # R. J. Mathar, Jan 07 2011

CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
(* also *)
t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i == 0, 0, True, b[n, i-1]+b[n-i*(i+1)/2, i]]; a[n_] := b[n, Floor[Sqrt[2*n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[Append[#,0]]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)
nmax = 58; t = Table[PolygonalNumber[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[t, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

N=66; Vec(1/prod(k=1,N,1-x^(k*(k+1)\2))+O(x^N)) \\ Joerg Arndt, Apr 14 2013

from functools import lru_cache
from sympy import divisors
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A007294(n):
    @lru_cache(maxsize=None)
    def a(n): return is_square((n<<3)+1)
    @lru_cache(maxsize=None)
    def c(n): return sum(d for d in divisors(n,generator=True) if a(d))
    return (c(n)+sum(c(k)*A007294(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

def A007294(n):
    has_nondecreasing_diffs = lambda x: min(differences(x, 2)) >= 0
    special = lambda x: (x[1]-x[0]) >= x[0]
    allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_nondecreasing_diffs(x))
    return len([1 for x in Partitions(n, max_slope=-1) if allowed(x[::-1])]) # D. S. McNeil, Jan 06 2011

G.f.: 1/Product_{k>=2} (1-z^binomial(k, 2)).
For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller, Aug 26 2003
For n>0, a(n) is Euler Transform of [1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...], i.e A010054, n>0. - Benedict W. J. Irwin, Jul 29 2016
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2) / (2^(7/2) * sqrt(3) * Pi * n^(3/2)) [Brigham 1950 (exponential part), Almkvist 2006]. - Vaclav Kotesovec, Dec 31 2016
G.f.: Sum_{i>=0} x^(i*(i+1)/2) / Product_{j=1..i} (1 - x^(j*(j+1)/2)). - Ilya Gutkovskiy, May 07 2017

Additional comments from Roland Bacher, Jun 17 2001

A007862 Number of triangular numbers that divide n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 5, 1, 1, 2, 1, 1, 4, 1, 1, 2, 2, 1, 4, 1, 1, 4, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 5, 1, 1, 3, 1, 1, 4, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 5, 1, 1, 2, 1, 1, 6, 2, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 5

Offset: 1

Richard Stanley

nonn

Also a(n) is the total number of ways to represent n+1 as a centered polygonal number of the form km(m+1)/2+1 for k>1. - Alexander Adamchuk, Apr 26 2007
Number of oblong numbers that divide 2n. - Ray Chandler, Jun 24 2008
The number of divisors d of 2n such that d+1 is also a divisor of 2n, see first formula. - Michel Marcus, Jun 18 2015
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n forming a finite arithmetic progression with offset 0, i.e. the differences are all equal (with the last part taken to be 0). The Heinz numbers of these partitions are given by A325327. For example, the a(1) = 1 through a(12) = 3 partitions are (A = 10, B = 11, C = 12):
  1    2    3     4    5    6      7    8    9     A       B     C
            21              42               63    4321          84
                            321                                  642
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A046951, A130317.
Cf. A010054, A027750, A000005, A239930.
Cf. A000217, A007294, A049988, A325324, A325327, A325407.

a007862 = sum . map a010054 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014

sup=90; TriN=Array[ (#+1)(#+2)/2&, Floor[ N[ Sqrt[ sup*2 ] ] ]-1 ]; Array[ Function[n, 1+Count[ Map[ Mod[ n, # ]&, TriN ], 0 ] ], sup ]
Table[Count[Divisors[k], ?(IntegerQ[Sqrt[8 # + 1]] &)], {k, 105}] (* _Jayanta Basu, Aug 12 2013 *)
Table[Length[Select[IntegerPartitions[n],SameQ@@Differences[Append[#,0]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ Michel Marcus, Jun 18 2015

from itertools import pairwise
from sympy import divisors
def A007862(n): return sum(1 for a, b in pairwise(divisors(n<<1)) if a+1==b)  # Chai Wah Wu, Jun 09 2025

a(n) = Sum_{d|2*n,d+1|2*n} 1.
G.f.: Sum_{k>=1} x^A000217(k)/(1-x^A000217(k)). - Jon Perry, Jul 03 2004
a(A130317(n)) = n and a(m) <> n for m < A130317(n). - Reinhard Zumkeller, May 23 2007
a(n) = A129308(2n). - Ray Chandler, Jun 24 2008
a(n) = Sum_{k=1..A000005(n)} A010054(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023

Extended by Ray Chandler, Jun 24 2008

A217605 Number of partitions of n that are fixed points of a certain map (see comment).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 3, 0, 3, 3, 3, 0, 4, 3, 2, 1, 6, 4, 5, 2, 5, 7, 10, 2, 10, 10, 11, 4, 9, 5, 14, 7, 13, 13, 18, 7, 20, 17, 22, 10, 22, 19, 32, 15, 26, 26, 40, 15, 37, 36, 43, 21, 44, 32, 55, 30, 46, 43, 75, 32, 67, 62, 83, 40, 82, 61, 104, 58, 89, 71, 136, 66, 114, 97, 149, 77, 143, 106, 176, 101, 160, 123, 222, 114, 190

Offset: 0

Joerg Arndt, Oct 08 2012

nonn

Writing a partition of n in the form sum(k>=1, c(k) * k) another (in general different) partition is obtained as sum(k>=1, k * c(k)).  For example, the partition 6 =  4* 1 + 1* 2 = 1 + 1 + 1 + 1 + 2 is mapped to 1* 4 + 2 *1 = 2* 1 + 1* 4 = 2 + 2 + 4.  This sequence counts the fixed points of this map.
The map is not surjective. For example, all partitions into distinct parts are mapped to n* 1.
The map is an involution for partitions where the multiplicities of all parts are distinct (Wilf partitions, see A098859). If in addition the set of parts the same as the set of multiplicities then the partition is a fixed point.
The second part of the preceding comment is incorrect. For example, the partition (3,3,2,1,1,1) maps to (3,2,2,2,1,1) so is not a fixed point, even though the set of parts is identical to the set of multiplicities. - Gus Wiseman, May 04 2019

			a(16) = 4 because the following partitions of 16 are fixed points:
  4* 2 + 2* 4  =   2 + 2 + 2 + 2 + 4 + 4
  4* 4  =   4 + 4 + 4 + 4
  6* 1 + 2* 2 + 1* 6  =   1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 6
  8* 1 + 1* 8  =   1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 8
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(16) = 4 partitions are the following (empty columns not shown). The Heinz numbers of these partitions are given by A048768.
  1  22   221  3111  41111  333  3331    33222    33322   333221    4444
     211                         322111  4221111  332221  52211111  442222
                                 511111  6111111  333211  71111111  622111111
                                                                    811111111
(End)

James Allen Fill, Svante Janson, Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf, The Electronic Journal of Combinatorics, vol.19, no.2, 2012.
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

Cf. A033461, A048767, A048768, A098859, A320348, A325324, A325325.

winv[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];
Table[Length[Select[IntegerPartitions[n],winv[Times@@Prime/@#]==Times@@Prime/@#&]],{n,0,30}] (* Gus Wiseman, May 04 2019 *)

A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 9, 7, 13, 12, 13, 16, 22, 17, 28, 28, 31, 36, 50, 45, 63, 62, 74, 78, 102, 92, 123, 123, 146, 148, 191, 181, 228, 233, 280, 283, 348, 350, 420, 437, 518, 523, 616, 641, 727, 774, 884, 911, 1038, 1102, 1240, 1292, 1463, 1530, 1715, 1861, 2002

Offset: 1

Seiichi Manyama, Oct 11 2018

nonn

Also the number of integer partitions of n whose parts cover an initial interval of positive integers with distinct multiplicities. Also the number of integer partitions of n whose multiplicities cover an initial interval of positive integers and are distinct (see A048767 for a bijection). - Gus Wiseman, May 04 2019

			n = 9
[9]        *********  a_1 = 9.
           ooooooooo
------------------------------------
[8, 1]             *        a_2 = 1.
            *******o  a_1 - a_2 = 7.
            oooooooo
------------------------------------
[7, 2]            **        a_2 = 2.
             *****oo  a_1 - a_2 = 5.
             ooooooo
------------------------------------
[5, 4]          ****        a_2 = 4.
               *oooo  a_1 - a_2 = 1.
               ooooo
------------------------------------
a(9) = 4.
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(11) = 9 strict partitions with distinct differences (where the last part is taken to be 0) are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A325388.
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)
                 (31)  (32)  (51)  (43)  (53)  (54)  (64)   (65)
                       (41)        (52)  (62)  (72)  (73)   (74)
                                   (61)  (71)  (81)  (82)   (83)
                                                     (91)   (92)
                                                     (631)  (A1)
                                                            (632)
                                                            (641)
                                                            (731)
The a(1) = 1 through a(10) = 6 partitions covering an initial interval of positive integers with distinct multiplicities are the following. The Heinz numbers of these partitions are given by A325326.
  1  11  111  211   221    21111   2221     22211     22221      222211
              1111  2111   111111  22111    221111    2211111    322111
                    11111          211111   2111111   21111111   2221111
                                   1111111  11111111  111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(1) = 1 through a(10) = 6 partitions whose multiplicities cover an initial interval of positive integers and are distinct are the following (A = 10). The Heinz numbers of these partitions are given by A325337.
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)
                 (211)  (221)  (411)  (322)  (332)  (441)  (433)
                        (311)         (331)  (422)  (522)  (442)
                                      (511)  (611)  (711)  (622)
                                                           (811)
                                                           (322111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000009, A320347.

Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[Append[#,0]]&]],{n,30}] (* Gus Wiseman, May 04 2019 *)

A325367 Heinz numbers of integer partitions with distinct differences between successive parts (with the last part taken to be zero).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97

Offset: 1

Gus Wiseman, May 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325324.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  28: {1,1,4}

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

Positions of squarefree numbers in A325390.

Cf. A056239, A112798, A130091, A320348, A325324, A325327, A325362, A325364, A325366, A325368, A325388, A325405, A325407, A325460, A325461.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[200],UnsameQ@@Differences[Append[primeptn[#],0]]&]

A325349 Number of integer partitions of n whose augmented differences are distinct.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 5, 7, 7, 12, 10, 13, 15, 21, 21, 31, 34, 38, 45, 55, 60, 71, 80, 84, 103, 119, 134, 152, 186, 192, 228, 263, 292, 321, 377, 399, 454, 514, 565, 618, 709, 752, 840, 958, 1050, 1140, 1297, 1402, 1568, 1755, 1901, 2080, 2343, 2524, 2758, 3074

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325366.

			The a(1) = 1 through a(11) = 10 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (22)  (41)  (33)  (43)   (44)   (54)   (55)   (65)
                  (31)        (42)  (52)   (62)   (63)   (64)   (83)
                              (51)  (61)   (71)   (72)   (73)   (92)
                                    (421)  (422)  (81)   (82)   (A1)
                                           (431)  (522)  (91)   (443)
                                           (521)  (621)  (433)  (641)
                                                         (442)  (722)
                                                         (541)  (731)
                                                         (622)  (821)
                                                         (631)
                                                         (721)
For example, (4,4,3) has augmented differences (1,2,3), which are distinct, so (4,4,3) is counted under a(11).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..440

Cf. A000837, A049988, A098859, A325324, A325325, A325328, A325351, A325366, A325404.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[Append[#,1]]&]],{n,0,30}]

A383512 Heinz numbers of conjugate Wilf partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
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.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
     1: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

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

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A383513 Heinz numbers of non conjugate Wilf partitions.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A381433 in having 65.
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.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}

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

Partitions of this type are counted by A336866.
The conjugate version is A130092, complement A130091.
Including differences of 0 gives complement of A325367, counted by A325324.
The strict case is the complement of A325388, counted by A320348.
The complement is A383512, counted by A098859.
Also forbidding distinct multiplicities gives A383531, counted by A383530.
These are positions of non-strict rows in A383534, or nonsquarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A238745, A325325, A325351, A325355, A325366, A325368, A381431, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 7, 7, 8, 11, 11, 14, 17, 19, 20, 27, 27, 35, 38, 45, 47, 60, 63, 75, 84, 97, 104, 127, 134, 155, 175, 196, 218, 251, 272, 307, 346, 384, 424, 480, 526, 586, 658, 719, 798, 890, 979, 1078, 1201, 1315, 1451, 1603, 1762, 1934, 2137

Offset: 0

Gus Wiseman, Jun 13 2025

			The strict partition y = (7,6,5,3,2,1) has maximal runs ((7,6,5),(3,2,1)), with lengths (3,3), so y is counted under a(24).
The a(1) = 1 through a(14) = 14 partitions (A-E = 10-14):
  1  2  3   4   5   6    7   8   9    A     B    C     D    E
        21  31  32  42   43  53  54   64    65   75    76   86
                41  51   52  62  63   73    74   84    85   95
                    321  61  71  72   82    83   93    94   A4
                                 81   91    92   A2    A3   B3
                                 432  631   A1   B1    B2   C2
                                 531  4321  641  543   C1   D1
                                            731  642   742  752
                                                 741   751  842
                                                 831   841  851
                                                 5421  931  941
                                                            A31
                                                            5432
                                                            6521

John Tyler Rascoe, Table of n, a(n) for n = 0..100

For subsets instead of strict partitions we have A243815, distinct lengths A384175.
For distinct instead of equal lengths we have A384178, for anti-runs A384880.
This is the strict case of A384904, distinct lengths A384884.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
Cf. A000217, A008284, A044813, A047966, A089259, A325324, A325325, A329739, A382857, A383013, A383708, A384176.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,15}]

A_q(N) = {Vec(1+sum(k=1,floor(-1/2+sqrt(2+2*N)), sum(i=1,(N/(k*(k+1)/2))+1, q^(k*(k+1)*i^2/2)/prod(j=1,i, 1 - q^(j*k)))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 21 2025

G.f.: 1 + Sum_{i,k>0} q^(k*(k+1)*i^2/2)/Product_{j=1..i} (1 - q^(j*k)). - John Tyler Rascoe, Aug 21 2025

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A325349 Number of integer partitions of n whose augmented differences are distinct.

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