A008289 -id:A008289 - cp's OEIS frontend

A365379 Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.

0, 1, 3, 5, 10, 14, 27, 35, 61, 83, 128, 166, 264, 327, 482, 632, 882, 1110, 1565, 1938, 2663, 3339, 4401, 5471, 7290, 8921, 11555, 14291, 18280, 22303, 28507, 34507, 43534, 52882, 65798, 79621, 98932, 118629, 146072, 175562, 214708, 256351, 312583, 371779

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

			The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).
The a(1) = 1 through a(5) = 14 partitions:
  (1)  (1)   (1)    (1)     (1)
       (2)   (3)    (2)     (5)
       (11)  (11)   (4)     (11)
             (21)   (11)    (21)
             (111)  (21)    (31)
                    (22)    (32)
                    (31)    (41)
                    (111)   (111)
                    (211)   (211)
                    (1111)  (221)
                            (311)
                            (1111)
                            (2111)
                            (11111)

For subsets with positive coefficients we have A088314, complement A088528.
The case of strict partitions with positive coefficients is also A088314.
The version for subsets is A365073, complement A365380.
The case of strict partitions is A365311, complement A365312.
The complement is counted by A365378.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n],combs[n,Union[#]]!={}&]],{n,0,10}]

from sympy.utilities.iterables import partitions
def A365379(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(21)-a(43) from Chai Wah Wu, Sep 13 2023

A147542 Product(1 + a(n)x^n, n=1..infinity) = sum(F(k+1)x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 4, 18, 8, 8, 18, 17, 40, 50, 88, 396, 210, 296, 492, 690, 1144, 1776, 2786, 3545, 6704, 10610, 16096, 25524, 39650, 63544, 97108, 269154, 236880, 389400, 589298, 956000, 1459960, 2393538, 3604880, 5739132, 9030450, 14777200

Offset: 1

Neil Fernandez, Nov 06 2008

nonn

A formal infinite product representation for the Fibonacci numbers (A000045(n+1)).

For references see A147541. [R. J. Mathar, Mar 12 2009]

Jean-François Alcover, Table of n, a(n) for n = 1..200
Wolfdieter Lang Two recurrences for the general problem.
R. J. Mathar, Re: polynomial-to-product transform, Maple code (2008). [From _R. J. Mathar_, Mar 12 2009]

Cf. A000045, A137852, A006973, A157159.

m = 200;
sol = Thread[CoefficientList[Sum[Log[1 + a[n] x^n], {n, 1, m}] - Log[1/(1 - x - x^2)] + O[x]^(m + 1), x] == 0] // Solve // First;
Array[a, m] /. sol (* Jean-François Alcover, Oct 22 2019 *)

From Wolfdieter Lang, Mar 06 2009: (Start)
Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)):
a(n)= F(n+1) - sum(sum(product(a(k[j]),j=1..m),fp from FP(n,m)),m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=F(2)=1, a(2)=F(3)=2. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With the definition of FP(n,m) from the above recurrence I, P(n,m) the general set of partitions of n with m parts, and the multinomial numbers M_0 (given for every partition under A048996):
a(n) = sum((d/n)*(-a(d)^(n/d)),d|n with 1=2; a(1)=F(2)=1. The exponents e(j)>=0 satisfy sum(j*e(j),j=1..n)=n and sum(e(j),j=1..m). The M_0 numbers are m!/product(e(j)!,j=1..n).
Example of recurrence I: a(4) = F(5) - a(1)*a(3) = 5 - 1*1 = 4.
Example of recurrence II: a(4)= 2*(-1)^2 + (1*F(5)-(1/2)*(2*F(2)*F(4) + 1*F(3)^2) + (1/3)*3*F(2)^2*F(3)) = 4. (End)

More terms and revised description from Wolfdieter Lang Mar 06 2009
Edited by N. J. A. Sloane, Mar 11 2009 at the suggestion of Vladeta Jovovic
More terms from R. J. Mathar, Mar 12 2009

A157159 Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.

Original entry on oeis.org

1, 1, -1, 10, -16, 126, -526, 10312, -30024, 453840, -2805408, 45779328, -374664720, 7932770496, -67692115440, 2432120198016, -16610113920768, 437275706750208, -5110200130727808, 159305381515284480, -1931470594025607936, 63854116254680514048

Offset: 1

Wolfdieter Lang Mar 06 2009

sign

			Recurrence I: a(7) = 6! - (7*a(1)*a(6) + 21*a(2)*a(5) + 35*a(3)*a(4) + 105*a(1)*a(2)*a(4)) = 720 - (7*126 + 21*(-16) + 35*(-1)*10 + 105*10) = -526.
Recurrence II: a(4) = 3!*(1+2*(-1/2!)^2) + 1 = +10.

Alois P. Heinz, Table of n, a(n) for n = 1..170

Cf. A137852, A006973, A147542.

a:= proc(n) option remember; `if`(n=1, 1, (n-1)!*((-1)^n+add(d*
      (-a(d)/d!)^(n/d), d=numtheory[divisors](n) minus {1, n}))
       +(-1)^(n+1)*add((k-1)!*Stirling1(n, k), k=1..n))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 14 2012

a[n_] := a[n] = If[n == 1, 1, (n-1)!*((-1)^n+Sum[d*(-a[d]/d!)^(n/d), {d, Divisors[n][[2 ;; -2]]}])+(-1)^(n+1)*Sum[(k-1)!*StirlingS1[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Definition of a(n): 1-log(1-x) = product(1+a(n)*(x^n)/n!, n=1..infinity) (formal series and product).
Recurrence I. With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions (fp)) and the multinomial numbers M1(fp(n,m)) (given as array in A036038 for any partition) for fp(n,m) from FP(n,m): a(n) = (n-1)! - sum(sum(M1(fp)*product(a(k[j]),j=1..m),fp from FP(n,m)), m=2..maxm(n)), with maxm(n):=A003056(n) and the distinct parts k[j], j=1,...,m, of the partition fp of n, n>=3. Inputs a(1)=1, a(2)=1. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: a(n) = (n-1)!*((-1)^n + sum(d*(-a(d)/d!)^(n/d),d|n with 1A089064(n), n>=2, a(1)=1. A089064(n)=sum(((-1)^(m-1))*(m-1)!)*|S1(n,m)|, m=1..n) with the unsigned Stirling numbers of the first kind |A008275|. See the W. Lang link under A147542 for these recurrences.

More terms from Alois P. Heinz, Aug 14 2012

A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 17, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The first element not in A326715 that is however a Heinz number of these partitions is 273.

			The a(n) partitions for n = 6, 12, 24, 90, 84:
  6       12        24            90                      84
  3,2,1   6,4,2     12,8,4        45,30,15                42,28,14
          6,3,2,1   12,6,4,2      45,30,9,5,1             42,21,14,7
                    12,8,3,1      45,18,15,9,3            42,28,12,2
                    8,6,4,3,2,1   45,30,10,3,2            42,28,6,4,3,1
                                  45,18,15,10,2           42,28,7,4,2,1
                                  45,30,6,5,3,1           42,14,12,7,6,3
                                  45,30,9,3,2,1           42,21,12,4,3,2
                                  45,15,10,9,6,5          42,21,12,6,2,1
                                  45,18,10,9,5,3          42,21,14,4,2,1
                                  45,18,10,9,6,2          28,21,14,12,6,3
                                  45,18,15,6,5,1          28,21,14,12,7,2
                                  45,18,15,9,2,1          42,21,7,6,4,3,1
                                  30,18,15,10,6,5,3,2,1   42,14,12,7,4,3,2
                                                          42,14,12,7,6,2,1
                                                          28,21,14,12,4,3,2
                                                          28,21,14,12,6,2,1

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2519 terms from Antti Karttunen)
Antti Karttunen, Scheme program for computing this sequence

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A326842 (A326847).
A018818 = partitions using divisors (A326841).
A047993 = balanced partitions (A106529).
A067538 = partitions whose length/maximum divides sum (A316413/A326836).
A072233 = partitions by sum and length, with strict case A008289.
A102627 = strict partitions whose length divides sum.
A326850 = strict partitions whose maximum part divides sum.
A326851 = strict partitions w/ length and max dividing sum.
A340828 = strict partitions w/ length divisible by max.
A340829 = strict partitions w/ Heinz number divisible by sum.
A340830 = strict partitions w/ parts divisible by length.
Cf. A064173, A143773 (A316428), A168659, A326843 (A326837), A326852, A330950 (A324851), A340851, A340853.

Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],UnsameQ@@#&&Divisible[n,Length[#]]&]],{n,30}]

A340827(n, divsleft=List(divisors(n)), rest=n, len=0) = if(rest<=0, !rest && !(n%len), my(s=0, d); forstep(i=#divsleft, 1, -1, d = divsleft[i]; listpop(divsleft,i); if(rest>=d, s += A340827(n, divsleft, rest-d, 1+len))); (s)); \\ Antti Karttunen, Feb 22 2023

;; See the Links-section. - Antti Karttunen, Feb 22 2023

Data section extended up to a(105) by Antti Karttunen, Feb 22 2023

A359896 Number of odd-length integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 6, 9, 11, 15, 27, 32, 50, 58, 72, 112, 149, 171, 246, 286, 359, 477, 630, 773, 941, 1181, 1418, 1749, 2289, 2668, 3429, 4162, 4878, 6074, 7091, 8590, 10834, 12891, 15180, 18491, 22314, 25845, 31657, 36394, 42269, 52547, 62414, 73576, 85701

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(4) = 1 through a(9) = 11 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (21111)  (331)    (422)      (522)
                         (421)    (431)      (621)
                         (511)    (521)      (711)
                         (22111)  (611)      (22221)
                         (31111)  (22211)    (32211)
                                  (32111)    (33111)
                                  (41111)    (42111)
                                  (2111111)  (51111)
                                             (2211111)
                                             (3111111)

These partitions are ranked by A359892.
The any-length version is A359894, complement A240219, strict A359898.
The complement is counted by A359895, ranked by A359891.
The strict case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A008289, A066571, A316313, A327472, A327482, A359890, A359906, A359910.

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&Mean[#]!=Median[#]&]],{n,0,30}]

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 6, 5, 11, 12, 14, 21, 29, 26, 44, 44, 58, 68, 92, 92, 118, 137, 165, 192, 241, 223, 324, 353, 405, 467, 518, 594, 741, 809, 911, 987, 1239, 1276, 1588, 1741, 1823, 2226, 2566, 2727, 3138, 3413, 3905, 4450, 5093, 5434, 6134

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(7) = 1 through a(13) = 11 partitions:
  (4,2,1)  (4,3,1)  (6,2,1)  (5,3,2)  (5,4,2)    (6,5,1)    (6,4,3)
           (5,2,1)           (5,4,1)  (6,3,2)    (7,3,2)    (6,5,2)
                             (6,3,1)  (6,4,1)    (8,3,1)    (7,4,2)
                             (7,2,1)  (7,3,1)    (9,2,1)    (7,5,1)
                                      (8,2,1)    (6,3,2,1)  (8,3,2)
                                      (5,3,2,1)             (8,4,1)
                                                            (9,3,1)
                                                            (10,2,1)
                                                            (5,4,3,1)
                                                            (6,4,2,1)
                                                            (7,3,2,1)

The non-strict version is ranked by A359890, complement A359889.
The non-strict version is A359894, complement A240219.
The complement is counted by A359897.
The odd-length case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A082550, A135342, A240851, A327475, A327482, A328966, A359906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Mean[#]!=Median[#]&]],{n,0,30}]

A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 50, 12, 14, 33, 54, 0, 115, 0, 92, 75, 31, 99, 323, 0, 45, 162, 443, 0, 507, 0, 467, 732, 88, 0, 1551, 274, 833, 627, 1228, 0, 2035, 1556, 2859, 1152, 221, 0, 9008, 0, 295, 4835, 5358

Offset: 1

Gus Wiseman, Jun 23 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(8) = 1 through a(18) = 12 partitions:
  3221  .  32221  .  4332    .  3222221  43332  5443      .  433332
                     5331       3322211  53331  6442         443331
                     322221     4222211  63321  7441         533322
                     422211                     32222221     533331
                                                33222211     543321
                                                42222211     633321
                                                52222111     733311
                                                             322222221
                                                             332222211
                                                             422222211
                                                             432222111
                                                             522222111

Non-constant partitions are counted by A144300, ranks A024619.
This is the non-constant case of A363719, ranks A363727.
These partitions have ranks A363729.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A363720, A363721.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A364673 Number of (necessarily strict) integer partitions of n containing all of their own first differences.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 5, 2, 2, 4, 2, 3, 6, 4, 4, 8, 4, 4, 10, 8, 7, 8, 13, 9, 15, 12, 13, 17, 20, 15, 31, 24, 27, 32, 33, 32, 50, 42, 45, 53, 61, 61, 85, 76, 86, 101, 108, 118, 137, 141, 147, 179, 184, 196, 222, 244, 257, 295, 324, 348, 380, 433

Offset: 0

Gus Wiseman, Aug 03 2023

nonn

			The partition y = (12,6,3,2,1) has differences (6,3,1,1), and {1,3,6} is a subset of {1,2,3,6,12}, so y is counted under a(24).
The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:
  (1)  (3)    (6)      (12)       (15)         (18)         (21)
       (2,1)  (4,2)    (8,4)      (10,5)       (12,6)       (14,7)
              (3,2,1)  (6,4,2)    (8,4,2,1)    (9,6,3)      (12,6,3)
                       (5,4,2,1)  (5,4,3,2,1)  (6,5,4,2,1)  (8,6,4,2,1)
                       (6,3,2,1)               (7,5,3,2,1)  (9,5,4,2,1)
                                               (8,4,3,2,1)  (9,6,3,2,1)
                                                            (10,5,3,2,1)
                                                            (6,5,4,3,2,1)

John Tyler Rascoe, Table of n, a(n) for n = 0..200

Containing all differences: A007862.
Containing no differences: A364464, strict complement A364536.
Containing at least one difference: A364467, complement A363260.
For subsets of {1..n} we have A364671, complement A364672.
A non-strict version is A364674.
For submultisets instead of subsets we have A364675.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A196723, A229816, A237667, A320347, A363225, A364272, A364345, A364463, A364537, A370386.

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

from collections import Counter
def A364673_list(maxn):
    count = Counter()
    for i in range(maxn//3):
        A,f,i = [[(i+1, )]],0,0
        while f == 0:
            A.append([])
            for j in A[i]:
                for k in j:
                    x = j + (j[-1] + k, )
                    y = sum(x)
                    if y <= maxn:
                        A[i+1].append(x)
                        count.update({y})
            if len(A[i+1]) < 1: f += 1
            i += 1
    return [count[z]+1 for z in range(maxn+1)] # John Tyler Rascoe, Mar 09 2024

A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 8, 4, 11, 9, 16, 14, 25, 20, 37, 31, 49, 47, 73, 64, 101, 96, 135, 133, 190, 181, 256, 253, 336, 342, 453, 452, 596, 609, 771, 803, 1014, 1041, 1309, 1362, 1674, 1760, 2151, 2249, 2736, 2884, 3449, 3661, 4366, 4615, 5486, 5825

Offset: 0

Gus Wiseman, Aug 31 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

			The a(8) = 2 through a(13) = 11 partitions:
  (8)    (9)      (10)       (11)       (12)       (13)
  (5,3)  (5,4)    (6,4)      (6,5)      (7,5)      (7,6)
         (7,2)    (7,3)      (7,4)      (5,4,3)    (8,5)
         (4,3,2)  (4,3,2,1)  (8,3)      (5,4,2,1)  (9,4)
                             (9,2)                 (10,3)
                             (5,4,2)               (11,2)
                             (6,3,2)               (6,4,3)
                             (5,3,2,1)             (6,5,2)
                                                   (7,4,2)
                                                   (5,4,3,1)
                                                   (6,4,2,1)

Chai Wah Wu, Table of n, a(n) for n = 0..109

The nonnegative version for subsets appears to be A124506.
For sums instead of combinations we have A364349, binary A364533.
The nonnegative version is A364350, complement A364839.
For subsets instead of partitions we have A365044, complement A365043.
The non-strict version is A365072, nonnegative A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364912 counts linear combinations of partitions of k.
Cf. A151897, A237113, A236912, A237667, A275972, A363226, A364272, A364913, A365004, A365068.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combp[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,30}]

from sympy.utilities.iterables import partitions
def A365006(n):
    if n <= 1: return 1
    alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
    c = 1
    for p in partitions(n,k=n-1):
        if max(p.values()) == 1:
            s = set(p)
            for q in s:
                if tuple(sorted(s-{q})) in alist[q]:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(56) from Chai Wah Wu, Sep 20 2023

A383090 Number of integer partitions of n having more than one permutation with all equal run-lengths.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 20, 28, 43, 55, 77, 107, 141, 183, 244, 312, 411, 521, 664, 837, 1069, 1328, 1667, 2069, 2578, 3166, 3929, 4791, 5895, 7168, 8749, 10594, 12883, 15500, 18741, 22493, 27069, 32334, 38760, 46133, 55065, 65367, 77686, 91905, 108927, 128431, 151674

Offset: 0

Gus Wiseman, Apr 19 2025

nonn

			The partition (3322221) has 3 permutations with all equal run-lengths: (2323212), (2321232), (2123232), so is counted under a(15).
The partition (3322111111) has 2 permutations with all equal run-lengths: (1133112211), (1122113311), so is counted under a(16).
The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (32211)
                                             (42111)
                                             (222111)

For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
Partitions of this type are ranked by A383089 = positions of terms > 1 in A382857.
The complement is A383091, counted by A383092.
For a unique choice we have A383094, ranks A383112.
The complement for run-sums is A383095 + A383096, ranks A383099 \/ A383100.
For run-sums we have A383097, ranked by A383015 = positions of terms > 1 in A382877.
For distinct instead of equal run-lengths we have A383111, ranks A383113.
Cf. A047966, A072774, A098859, A304442, A100471, A100881, A100882, A100883.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A047993, A237984, A329739, A381541, A381871, A382076, A382771.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]

The complement is counted by A383094 + A382915, ranks A383112 \/ A382879.

More terms from Bert Dobbelaere, Apr 26 2025

cp's OEIS Frontend

A365379 Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.

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A147542 Product(1 + a(n)*x^n, n=1..infinity) = sum(F(k+1)*x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).

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A157159 Infinite product representation of series 1 - log(1-x) = 1 + Sum_{j>=1} (j-1)!*(x^j)/j!.

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Maple

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A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

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A359896 Number of odd-length integer partitions of n whose parts do not have the same mean as median.

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A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

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A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A364673 Number of (necessarily strict) integer partitions of n containing all of their own first differences.

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A147542 Product(1 + a(n)x^n, n=1..infinity) = sum(F(k+1)x^k, k=1..infinity) = 1/(1-x-x^2), where F(n) = A000045(n) (Fibonacci numbers).