A358913 -id:A358913 - 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

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!.

cp's OEIS Frontend

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

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