A279787 -id:A279787 - cp's OEIS frontend

A358831 Number of twice-partitions of n into partitions with weakly decreasing lengths.

1, 1, 3, 6, 14, 26, 56, 102, 205, 372, 708, 1260, 2345, 4100, 7388, 12819, 22603, 38658, 67108, 113465, 193876, 324980, 547640, 909044, 1516609, 2495023, 4118211, 6726997, 11002924, 17836022, 28948687, 46604803, 75074397, 120134298, 192188760, 305709858, 486140940

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(4) = 14 twice-partitions:
  (1)  (2)     (3)        (4)
       (11)    (21)       (22)
       (1)(1)  (111)      (31)
               (2)(1)     (211)
               (11)(1)    (1111)
               (1)(1)(1)  (2)(2)
                          (3)(1)
                          (11)(2)
                          (21)(1)
                          (11)(11)
                          (111)(1)
                          (2)(1)(1)
                          (11)(1)(1)
                          (1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..500

This is the semi-ordered case of A141199.
For constant instead of weakly decreasing lengths we have A306319.
For distinct instead of weakly decreasing lengths we have A358830.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A196545 counts p-trees, enriched A289501.
Cf. A000041, A000219, A001970, A061260, A072233, A271619, A279787, A358836.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],GreaterEqual@@Length/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=Vec(P(n,y)-1), v=[1]); for(k=1, n, my(p=g[k], u=v); v=vector(k+1); v[1] = 1 + O(x*x^n); for(j=1, k, v[1+j] = (v[j] + if(jAndrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Original entry on oeis.org

1, 3, 5, 13, 17, 57, 65, 209, 321, 801, 1025, 3905, 4097, 12417, 21505, 53505, 65537, 233985, 262145, 885761, 1327105, 3147777, 4194305, 16060417, 17825793, 50339841, 84148225, 220217345, 268435457, 990937089, 1073741825, 3506503681

Offset: 1

Miklos Kristof, Sep 11 2002

A034729 = Sum_{d|n} (2^(d-1)).
A055895 = 2*A034729.
If p is a prime, then a(p) = A034729(p) = 2^(p-1)+1.
From Gus Wiseman, Jul 14 2020: (Start)
Number of ways to tile a rectangle of size n using horizontal strips. Also the number of ways to choose a composition of each part of a constant partition of n. The a(0) = 1 through a(5) = 17 splittings are:
  ()  (1)  (2)      (3)          (4)              (5)
           (1,1)    (1,2)        (1,3)            (1,4)
           (1),(1)  (2,1)        (2,2)            (2,3)
                    (1,1,1)      (3,1)            (3,2)
                    (1),(1),(1)  (1,1,2)          (4,1)
                                 (1,2,1)          (1,1,3)
                                 (2,1,1)          (1,2,2)
                                 (2),(2)          (1,3,1)
                                 (1,1,1,1)        (2,1,2)
                                 (1,1),(2)        (2,2,1)
                                 (2),(1,1)        (3,1,1)
                                 (1,1),(1,1)      (1,1,1,2)
                                 (1),(1),(1),(1)  (1,1,2,1)
                                                  (1,2,1,1)
                                                  (2,1,1,1)
                                                  (1,1,1,1,1)
                                                  (1),(1),(1),(1),(1)
(End)

			Divisors of 6 = 1,2,3,6 and 6-1 = 5, 6-2 = 4, 6-3 = 3, 6-6 = 0. a(6) = 2^5 + 2^4 + 2^3 + 2^0 = 32 + 16 + 8 + 1 = 57.
G.f. = x + 3*x^2 + 5*x^3 + 13*x^4 + 17*x^5 + 57*x^6 + 65*x^7 + ...
a(14) = 1 + 2^7 + 2^12 + 2^13 = 12417. - _Gus Wiseman_, Jun 20 2018

Gus Wiseman, Table of n, a(n) for n = 1..32

Cf. A055895, A034729.
Cf. A080267.
Cf. A051731.
The version looking at lengths instead of sums is A101509.
The strictly increasing (or strictly decreasing) version is A304961.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Requiring distinct instead of equal sums gives A336127.
Starting with a strict composition gives A336130.
Partitions of partitions are A001970.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Cf. A006951, A063834, A075900, A279787, A305551, A316245, A323433, A336128.

a[ n_] := If[ n < 1, 0, Sum[ 2^(n - d), {d, Divisors[n]}]] (* Michael Somos, Mar 28 2013 *)

a(n)=if(n<1,0,2^n*polcoeff(sum(k=1,n,2/(2-x^k),x*O(x^n)),n))

a(n) = sumdiv(n,d, 2^(n-d) ); /* Joerg Arndt, Mar 28 2013 */

G.f.: 2^n times coefficient of x^n in Sum_{k>=1} x^k/(2-x^k). - Benoit Cloitre, Apr 21 2003; corrected by Joerg Arndt, Mar 28 2013
G.f.: Sum_{k>0} 2^(k-1)*x^k/(1-2^(k-1)*x^k). - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=2*x and a=1/2. - Joerg Arndt, Jan 30 2011
Triangle A051731 mod 2 converted to decimal. - Philippe Deléham, Oct 04 2003
G.f.: Sum_{k>0} 1 / (2 / (2*x)^k - 1). - Michael Somos, Mar 28 2013

a(14) corrected from 9407 to 12417 by Gus Wiseman, Jun 20 2018

A279788 Twice partitioned numbers where the first partition is constant and the latter partitions are strict.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 10, 6, 12, 17, 21, 13, 57, 19, 49, 87, 86, 39, 240, 55, 279, 330, 235, 105, 1141, 386, 491, 1217, 1461, 257, 4804, 341, 2968, 4225, 1958, 5898, 18961, 761, 3782, 15007, 30572, 1261, 66245, 1611, 32523, 106951, 13122, 2591, 283013, 81390, 182873

Offset: 0

Gus Wiseman, Dec 18 2016

nonn

			The a(6)=10 twice-partitions are:
((6)), ((51)), ((42)), ((3)(3)), ((3)(21)), ((21)(3)),
((321)), ((2)(2)(2)), ((21)(21)), ((1)(1)(1)(1)(1)(1)).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf. A000005, A000009, A018818, A063834, A279787.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(n/d)^d, d=divisors(n)))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016

Table[DivisorSum[n,PartitionsQ[n/#]^#&],{n,20}]

A306319 Number of length-rectangular twice-partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 60, 82, 131, 177, 286, 376, 582, 793, 1202, 1610, 2450, 3274, 4906, 6665, 9770, 13274, 19690, 26506, 38596, 53006, 76432, 104189, 150844, 205282, 294304, 404146, 573140, 786169, 1119457, 1527554, 2155953, 2965567, 4163955, 5701816

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is length-rectangular if all parts have the same number of parts.

			The a(5) = 14 length-rectangular twice-partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Dominates A319066 (rectangular partitions of partitions), which dominates A323429 (rectangular plane partitions).

Cf. A000219, A001970, A063834 (twice-partitions), A089299, A271619, A279787 (sum-rectangular twice-partitions), A305551, A306017, A306318 (square case), A323531.

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],SameQ@@Length/@#&],{ptn,IntegerPartitions[n]}]],{n,20}]

A336137 Number of set partitions of the binary indices of n with equal block-sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(n) set partitions for n = 7, 59, 119, 367, 127:
  {123}    {12456}      {123567}      {1234679}    {1234567}
  {12}{3}  {126}{45}    {1236}{57}    {12346}{79}  {1247}{356}
           {15}{24}{6}  {156}{237}    {1249}{367}  {1256}{347}
                        {17}{26}{35}  {1267}{349}  {1346}{257}
                                      {169}{2347}  {167}{2345}
                                                   {16}{25}{34}{7}
The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
  {{2,7},{3,6},{4,5},{9}}
  {{2,4,6},{3,9},{5,7}}
  {{2,7,9},{3,4,5,6}}
  {{2,3,4,9},{5,6,7}}
  {{2,3,6,7},{4,5,9}}
  {{2,4,5,7},{3,6,9}}
  {{2,3,4,5,6,7,9}}
so a(382) = 7.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

These set partitions are counted by A035470.
The version for twice-partitions is A279787.
The version for partitions of partitions is A305551.
The version for factorizations is A321455.
The version for normal multiset partitions is A326518.
The version for distinct block-sums is A336138.
Set partitions of binary indices are A050315.
Normal multiset partitions with equal lengths are A317583.
Normal multiset partitions with equal averages are A326520.
Multiset partitions with equal block-sums are ranked by A326534.
Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]

A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 49, 91, 179, 341, 664, 1280, 2503, 4872, 9557, 18750, 36927, 72800, 143880, 284660, 564093, 1118911, 2221834, 4415417, 8781591, 17476099, 34799199, 69327512, 138176461, 275503854, 549502119, 1096327380, 2187894634, 4367310138, 8719509111

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(0) = 1 through a(4) = 13 sequences:
  ()  ((1))  ((2))     ((3))        ((4))
             ((11))    ((21))       ((22))
             ((1)(1))  ((111))      ((31))
                       ((1)(2))     ((211))
                       ((2)(1))     ((1111))
                       ((1)(1)(1))  ((1)(3))
                                    ((2)(2))
                                    ((3)(1))
                                    ((11)(11))
                                    ((1)(1)(2))
                                    ((1)(2)(1))
                                    ((2)(1)(1))
                                    ((1)(1)(1)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of set partitions is A038041.
The version for weakly decreasing lengths is A141199, strictly A358836.
For equal sums instead of lengths we have A279787.
The case of twice-partitions is A306319, distinct A358830.
The unordered version is A319066.
The case of plane partitions is A323429.
The case of constant sums also is A358833.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A000219, A001970, A063834, A218482, A305551, A358835.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],SameQ@@Length/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(1 + sum(k=1, n, 1/(1 - polcoef(g, k, y)) - 1))} \\ Andrew Howroyd, Dec 31 2022

G.f.: 1 + Sum_{k>=1} (1/(1 - [y^k]P(x,y)) - 1) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022

A358906 Number of finite sequences of distinct integer partitions with total sum n.

Original entry on oeis.org

1, 1, 2, 7, 13, 35, 87, 191, 470, 1080, 2532, 5778, 13569, 30715, 69583, 160386, 360709, 814597, 1824055, 4102430, 9158405, 20378692, 45215496, 100055269, 221388993, 486872610, 1069846372, 2343798452, 5127889666, 11186214519, 24351106180, 52896439646

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 13 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((1)(11))  ((1)(3))
                 ((11)(1))  ((3)(1))
                            ((11)(2))
                            ((1)(21))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

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

This is the case of A055887 with distinct partitions.
The unordered version is A261049.
The case of twice-partitions is A296122.
The case of distinct sums is A336342, constant sums A279787.
The version for sequences of compositions is A358907.
The case of weakly decreasing lengths is A358908.
The case of distinct lengths is A358912.
The version for strict partitions is A358913, distinct case of A304969.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all distinct Omegas.
Cf. A000009, A000041, A000219, A098407, A271619, A330463, A358836, A358901, A358914.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 13 2024

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@#&]],{n,0,10}]

a(n) = Sum_{k} A330463(n,k) * k!.

A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 7, 9, 12, 20, 21, 39, 49, 79, 109, 161, 236, 345, 512, 752, 1092, 1628, 2376, 3537, 5171, 7650, 11266, 16634, 24537, 36173, 53377, 78791, 116224, 171598, 253109, 373715, 551434, 814066, 1201466, 1773425, 2617744, 3864050, 5703840, 8419699

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

			The a(1) = 1 through a(8) = 9 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (23)     (33)      (25)       (35)
                    (1111)  (32)     (222)     (52)       (44)
                            (11111)  (111111)  (223)      (53)
                                               (232)      (233)
                                               (322)      (323)
                                               (1111111)  (332)
                                                          (2222)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

The case of partitions is A319169, ranked by A320324.
The weakly decreasing version is A358335, strictly A358901.
For sequences of partitions see A358905.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A358902 = compositions with weakly decreasing A001221, strictly A358903.
A358909 = partitions with weakly decreasing A001222, complement A358910.
Cf. A056239, A063834, A064573, A218482, A279787, A300335, A319066, A319071, A358831, A358908.

b:= proc(n, i) option remember; uses numtheory; `if`(n=0, 1, add(
     (t-> `if`(i<0 or i=t, b(n-j, t), 0))(bigomega(j)), j=1..n))
    end:
a:= n-> b(n, -1):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 12 2024

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,0,10}]

a(21) and beyond from Lucas A. Brown, Dec 15 2022

A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 3, 16, 8, 14, 3, 39, 3, 16, 15, 40, 3, 50, 3, 54, 17, 20, 3, 135, 10, 22, 25, 73, 3, 129, 3, 119, 21, 26, 19, 273, 3, 28, 23, 217, 3, 203, 3, 123, 74, 32, 3, 590, 12, 106, 27, 154, 3, 370, 23, 343, 29, 38, 3, 963, 3, 40, 95, 450, 25, 467, 3

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

An unlabeled version of A279789.

			The a(1) = 1 through a(6) = 12 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (111)      (22)          (11111)          (33)
       (1)(1)  (1)(1)(1)  (1111)        (1)(1)(1)(1)(1)  (222)
                          (2)(2)                         (3)(3)
                          (2)(11)                        (111111)
                          (11)(11)                       (3)(111)
                          (1)(1)(1)(1)                   (2)(2)(2)
                                                         (111)(111)
                                                         (2)(2)(11)
                                                         (2)(11)(11)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(1)(1)(1)

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A001970, A006171 (constant parts), A007716, A034729, A047966 (uniform partitions), A047968, A279787, A279789 (twice-partitions version), A305551 (equal part-sums), A306017, A319056, A323766, A323775, A323776.

Table[Length[Join@@Table[Union[Sort/@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@ptn]],{ptn,Select[IntegerPartitions[n],SameQ@@#&]}]],{n,30}]

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(d) + n/d - 1, n/d))); \\ Michel Marcus, Jan 28 2019

a(0) = 1; a(n) = Sum_{d|n} binomial(tau(d) + n/d - 1, n/d), where tau = A000005.

A295923 Number of twice-factorizations of n where the first factorization is constant, i.e., type (P,R,P).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 10, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 13, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 29, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 10, 2, 1, 11

Offset: 1

Gus Wiseman, Nov 30 2017

nonn

a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) factorizations of d.

			The a(16) = 10 twice-factorizations are (2*2*2*2), (2*2*4), (2*8), (4*4), (16), (2*2)*(2*2), (2*2)*(4), (4)*(2*2), (4)*(4), (2)*(2)*(2)*(2).

Cf. A000005, A001055, A052409, A052410, A279787, A281113, A284639, A295920, A295924, A295931, A295935.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
a[n_]:=Sum[Length[postfacs[n^(1/g)]]^g,{g,Divisors[GCD@@FactorInteger[n][[All,2]]]}];
Array[a,50]

cp's OEIS Frontend

A358831 Number of twice-partitions of n into partitions with weakly decreasing lengths.

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A074854 a(n) = Sum_{d|n} (2^(n-d)).

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A279788 Twice partitioned numbers where the first partition is constant and the latter partitions are strict.

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A306319 Number of length-rectangular twice-partitions of n.

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A336137 Number of set partitions of the binary indices of n with equal block-sums.

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Mathematica

A358905 Number of sequences of integer partitions with total sum n that are rectangular, meaning all lengths are equal.

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Mathematica

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A358906 Number of finite sequences of distinct integer partitions with total sum n.

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Maple

Mathematica

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A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.