A358901 -id:A358901 - cp's OEIS frontend

A358914 Number of twice-partitions of n into distinct strict partitions.

1, 1, 1, 3, 4, 7, 13, 20, 32, 51, 83, 130, 206, 320, 496, 759, 1171, 1786, 2714, 4104, 6193, 9286, 13920, 20737, 30865, 45721, 67632, 99683, 146604, 214865, 314782, 459136, 668867, 972425, 1410458, 2040894, 2950839, 4253713, 6123836, 8801349, 12627079

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(6) = 13 twice-partitions:
  ((1))  ((2))  ((3))     ((4))      ((5))      ((6))
                ((21))    ((31))     ((32))     ((42))
                ((2)(1))  ((3)(1))   ((41))     ((51))
                          ((21)(1))  ((3)(2))   ((321))
                                     ((4)(1))   ((4)(2))
                                     ((21)(2))  ((5)(1))
                                     ((31)(1))  ((21)(3))
                                                ((31)(2))
                                                ((3)(21))
                                                ((32)(1))
                                                ((41)(1))
                                                ((3)(2)(1))
                                                ((21)(2)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..100

The unordered version is A050342, non-strict A261049.
This is the distinct case of A270995.
The case of strictly decreasing sums is A279785.
The case of constant sums is A279791.
For distinct instead of weakly decreasing sums we have A336343.
This is the twice-partition case of A358913.
A001970 counts multiset partitions of integer partitions.
A055887 counts sequences of partitions.
A063834 counts twice-partitions.
A330462 counts set systems by total sum and length.
A358830 counts twice-partitions with distinct lengths.
Cf. A000009, A000219, A075900, A271619, A296122, A304969, A321449, A336342, A358901, A358906, A358907.

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

seq(n,k)={my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))-1)); Vec(prod(k=1, n, my(c=u[k]); sum(j=0, min(c,n\k), x^(j*k)*c!/(c-j)!,  O(x*x^n))))} \\ Andrew Howroyd, Dec 31 2022

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

A358908 Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.

Original entry on oeis.org

1, 1, 2, 6, 10, 23, 50, 95, 188, 378, 747, 1414, 2739, 5179, 9811, 18562, 34491, 64131, 118607, 218369, 400196, 731414, 1328069, 2406363, 4346152, 7819549, 14027500, 25090582, 44749372, 79586074, 141214698, 249882141, 441176493, 777107137, 1365801088, 2395427040, 4192702241

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

			The a(1) = 1 through a(4) = 10 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((11)(1))  ((1)(3))
                            ((3)(1))
                            ((11)(2))
                            ((21)(1))
                            ((111)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..200

This is the distinct case of A055887 with weakly decreasing lengths.
This is the distinct case is A141199.
The case of distinct lengths also is A358836.
This is the case of A358906 with weakly decreasing lengths.
A000041 counts integer partitions, strict A000009.
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.
A358912 counts sequences of partitions with distinct lengths.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000219, A261049, A271619, A296122, A358831, A358901, A358905, A358907.

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

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n,v) = {[subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, #v, (1 + y*x^k + O(x*x^n))^v[k] ))]}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, Ser(R(n, Vec(polcoef(g, k, y), -n)))  ))} \\ 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

A358907 Number of finite sequences of distinct integer compositions with total sum n.

Original entry on oeis.org

1, 1, 2, 8, 18, 54, 156, 412, 1168, 3200, 8848, 24192, 66632, 181912, 495536, 1354880, 3680352, 9997056, 27093216, 73376512, 198355840, 535319168, 1443042688, 3884515008, 10445579840, 28046885824, 75225974912, 201536064896, 539339293824, 1441781213952

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 18 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((12))     ((13))
                 ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((112))
                 ((2)(1))   ((121))
                 ((1)(11))  ((211))
                 ((11)(1))  ((1111))
                            ((1)(3))
                            ((3)(1))
                            ((1)(12))
                            ((11)(2))
                            ((1)(21))
                            ((12)(1))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

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

For sets instead of sequences we have A098407, partitions A261049.
This is the strict case of A133494.
The case of distinct sums is A336127, constant sums A074854.
The version for sequences of partitions is A358906.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A218482 counts sequences of compositions with weakly decreasing lengths.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all different Omegas.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.

g:= proc(n) option remember; ceil(2^(n-1)) end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
      add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 15 2022

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

a(16)-a(29) from Alois P. Heinz, Dec 15 2022

A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 7, 9, 10, 10, 10, 9, 11, 15, 14, 13, 15, 14, 14, 17, 16, 17, 17, 16, 16, 17, 17, 21, 26, 24, 23, 25, 27, 29, 32, 31, 29, 36, 36, 35, 37, 37, 42, 49, 45, 44, 50, 49, 50, 58, 55, 55, 58, 56, 58, 66, 62, 65, 75

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(15) = 8 partitions are: (15), (14,1), (12,3), (12,2,1), (10,5), (10,4,1), (6,9), (8,6,1).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

Counting prime factors with multiplicity gives A358901.
The weakly decreasing version is A358902, with multiplicity A358335.
A001222 counts prime factors, distinct A001221.
A116608 counts partitions by sum and number of distinct parts.
A358836 counts multiset partitions with all distinct block sizes.
Cf. A046660, A071625, A129519, A141199, A319169, A358831, A358909, A358911.

p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t b(n$2):
seq(a(n), n=0..68);  # Alois P. Heinz, Feb 14 2024

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

a(56) and beyond from Lucas A. Brown, Dec 14 2022

A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

Original entry on oeis.org

1, 1, 2, 5, 11, 23, 49, 103, 214, 434, 874, 1738, 3443, 6765, 13193, 25512, 48957, 93267, 176595, 332550, 622957, 1161230, 2153710, 3974809, 7299707, 13343290, 24280924, 43999100, 79412942, 142792535, 255826836, 456735456, 812627069, 1440971069, 2546729830

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

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

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

The case of set partitions is A007837.
This is the case of A055887 with all distinct lengths.
For distinct sums instead of lengths we have A336342.
The case of twice-partitions is A358830.
The unordered version is A358836.
The version for constant instead of distinct lengths is A358905.
A000041 counts integer partitions, strict A000009.
A063834 counts twice-partitions.
A141199 counts sequences of partitions with weakly decreasing lengths.
Cf. A000219, A001970, A038041, A060642, A218482, A271619, A319066, A358831, A358901, A358906, A358908.

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@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)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*polcoef(g, k, y) + O(x*x^n)))]} \\ Andrew Howroyd, Dec 30 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 30 2022

A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 68, 100, 153, 227, 342, 509, 759, 1129, 1678, 2492, 3699, 5477, 8121, 12015, 17795, 26313, 38924, 57541, 85065, 125712, 185758, 274431, 405420, 598815, 884465, 1306165, 1928943, 2848360, 4205979, 6210289, 9169540

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(0) = 1 through a(6) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (33)
                 (111)  (31)    (32)     (42)
                        (211)   (41)     (51)
                        (1111)  (221)    (222)
                                (311)    (231)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.

For lengths of partitions see A141199, compositions A218482.
The strictly decreasing case is A358901.
The version not counting multiplicity is A358902, strict A358903.
The case of partitions is A358909, complement A358910.
The case of equality is A358911, partitions A319169.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.

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

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

A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The case of compositions is A358335, strictly decreasing A358901.
The complement is counted by A358910.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Table[Length[Select[IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

cp's OEIS Frontend

A358914 Number of twice-partitions of n into distinct strict partitions.

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A358908 Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.

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

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A358907 Number of finite sequences of distinct integer compositions with total sum n.

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Maple

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A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

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Maple

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A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

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A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).