A318565 -id:A318565 - cp's OEIS frontend

A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

1, 1, 2, 7, 13, 34, 81, 175, 403, 890, 1977, 4262, 9356, 19963, 42573, 90865, 191206, 401803, 837898, 1744231, 3607504, 7436628, 15254309, 31185686, 63552725, 128963236, 260933000, 526140540, 1057927323, 2120500885, 4239012067, 8449746787, 16799938614

Offset: 0

Gus Wiseman, Dec 16 2019

nonn

Number of sets of disjoint nonempty sets of nonempty multisets of positive integers with total sum n.

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

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

Cf. A001970, A007713, A050343, A063834, A089259, A261049, A271619, A279375, A294617, A318565, A323787-A323795, A330452-A330459, A330460.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],UnsameQ@@Join@@#&]],{n,0,10}]

\\ here BellP is A000110 as series.
BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
seq(n)={my(b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ Andrew Howroyd, Dec 29 2019

a(n) = Sum_{0 <= k <= n} A330463(n,k) * A000110(k).

Terms a(18) and beyond from Andrew Howroyd, Dec 29 2019

A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

Original entry on oeis.org

1, 4, 10, 32, 80, 239, 605, 1670, 4251

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318563, A318564, A318565, A318566.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
strnormQ[m_]:=OrderedQ[Length/@Split[m],GreaterEqual];
Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@strnormQ/@#[[2]]&]],{n,6}]

A318563 Number of combinatory separations of strongly normal multisets of weight n.

Original entry on oeis.org

1, 4, 10, 33, 85, 272, 730, 2197, 6133

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318562, A318564, A318565, A318566.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}]],{n,7}]

A318567 Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.

Original entry on oeis.org

1, 3, 8, 21, 54, 137, 343, 847, 2075, 5031, 12109, 28921, 68633, 161865, 379655

Offset: 1

Gus Wiseman, Aug 29 2018

Also the number of combinatory separations of normal multisets of weight n with constant parts. A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}.

			The a(3) = 8 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={1,11}
  112<={1,1,1}
  122<={1,11}
  122<={1,1,1}
  123<={1,1,1}

Cf. A000041, A007716, A011782, A034691, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318560, A318562, A318565.

Table[Sum[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@c]]],{c,Join@@Permutations/@IntegerPartitions[n]}],{n,30}]

A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 4, 1, 6, 1, 1, 1, 1, 2, 6, 1, 5, 1, 7, 1, 1, 1, 1, 2, 3, 10, 1, 6, 1, 8, 1, 1, 1, 1, 1, 3, 4, 15, 1, 7, 1, 9, 1, 1, 1, 1, 4, 1, 4, 5, 21, 1, 8, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 25 2019

An orderless factorization of n with k > 1 levels of parentheses is any multiset partition of an orderless factorization of n with k - 1 levels of parentheses. If k = 1 it is just an orderless factorization of n into factors > 1.

			Array begins:
       k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12
   n=1: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=2: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=3: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=4: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=5: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=6: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=7: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=8: 1    3    6   10   15   21   28   36   45   55   66   78   91
   n=9: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=10: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=11: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=12: 1    4    9   16   25   36   49   64   81  100  121  144  169
  n=13: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=14: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=15: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=16: 1    5   14   30   55   91  140  204  285  385  506  650  819
  n=17: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=18: 1    4    9   16   25   36   49   64   81  100  121  144  169
The A(12,3) = 16 orderless factorizations of 12 with 2 levels of parentheses:
  ((2*2*3))          ((2*6))      ((3*4))      ((12))
  ((2)*(2*3))        ((2)*(6))    ((3)*(4))
  ((3)*(2*2))        ((2))*((6))  ((3))*((4))
  ((2))*((2*3))
  ((2)*(2)*(3))
  ((3))*((2*2))
  ((2))*((2)*(3))
  ((3))*((2)*(2))
  ((2))*((2))*((3))

Columns: A001055 (k=1), A050336 (k=2), A050338 (k=3), A050340 (k=4).
Rows: A000027, A000217, A000290, etc.
Cf. A096751, A141268, A144150, A213427, A255906, A281113, A290353, A292504, A317145, A318564, A318565, A318812, A323718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
lev[n_,k_]:=If[k==0,{n},Join@@Table[Union[Sort/@Tuples[lev[#,k-1]&/@fac]],{fac,facs[n]}]];
Table[Length[lev[sum-k,k]],{sum,12},{k,0,sum-1}]

A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 2, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 0, 2, 1, 3, 2, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 3, 2, 1

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The finite multiset of finite multisets of finite multisets of positive integers with MMM number n is obtained by factoring n into prime numbers, then factoring each of their prime indices into prime numbers, then factoring each of their prime indices into prime numbers, and finally taking their prime indices.

			The sequence of all finite multisets of finite multisets of finite multisets of positive integers begins (o is the empty multiset):
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((1)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((1)))
  11: (((2)))
  12: (oo(o))
  13: ((o(1)))
  14: (o(oo))
  15: ((o)((1)))
  16: (oooo)
  17: (((11)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((1)))

Cf. A000081, A000720, A001222, A050338, A056239, A112798, A301595, A302242, A318564, A318565, A318566, A324928.

fi[n_]:=If[n==1,{},FactorInteger[n]];
Table[Total[Cases[fi[n],{p_,k_}:>k*Total[Cases[fi[PrimePi[p]],{q_,j_}:>j*PrimeOmega[PrimePi[q]]]]]],{n,60}]

A318816 Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 3, 4, 3, 5, 14, 14, 9, 20, 9, 5, 14, 9, 5, 7, 28, 28, 33, 80, 33, 16, 68, 52, 16, 7, 28, 33, 16, 7, 11, 69, 69, 104, 266, 104, 74, 356, 282, 74, 29, 199, 253, 118, 29, 11, 69, 104, 74, 29, 11, 15, 134, 134, 294, 800, 294, 263, 1427, 1164

Offset: 1

Gus Wiseman, Sep 04 2018

			Tetrangle begins:
  1   2     3        5             7
      2 2   4 4     14 14         28 28
            3 4 3    9 20  9      33 80 33
                     5 14  9  5   16 68 52 16
                                   7 28 33 16  7
Non-isomorphic representatives of the T(4,3,2) = 20 multiset partitions:
  {{{1}},{{1},{1,1}}}  {{{1,1}},{{1},{1}}}
  {{{1}},{{1},{1,2}}}  {{{1,1}},{{1},{2}}}
  {{{1}},{{1},{2,2}}}  {{{1,1}},{{2},{2}}}
  {{{1}},{{1},{2,3}}}  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,1}}}  {{{1,2}},{{1},{1}}}
  {{{1}},{{2},{1,2}}}  {{{1,2}},{{1},{2}}}
  {{{1}},{{2},{1,3}}}  {{{1,2}},{{1},{3}}}
  {{{1}},{{2},{3,4}}}  {{{1,2}},{{3},{4}}}
  {{{2}},{{1},{1,1}}}  {{{2,3}},{{1},{1}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}

Cf. A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318393, A318399, A318564, A318565, A318566.

A330464 Number of non-isomorphic weight-n sets of set-systems with distinct multiset unions.

Original entry on oeis.org

1, 1, 3, 9, 32, 111, 463, 1942

Offset: 0

Gus Wiseman, Dec 26 2019

A set-system is a finite set of finite nonempty sets of positive integers.

As an alternative description, a(n) is the number of non-isomorphic sets of sets of sets with n leaves where the inner sets of sets all have different multiset unions.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 sets:
  {}  {{{1}}}  {{{1,2}}}      {{{1,2,3}}}
               {{{1},{2}}}    {{{1},{1,2}}}
               {{{1}},{{2}}}  {{{1},{2,3}}}
                              {{{1}},{{1,2}}}
                              {{{1}},{{2,3}}}
                              {{{1},{2},{3}}}
                              {{{1}},{{1},{2}}}
                              {{{1}},{{2},{3}}}
                              {{{1}},{{2}},{{3}}}

Non-isomorphic sets of sets are A283877.
Non-isomorphic sets of sets of sets are A323790.
Non-isomorphic set partitions of set-systems are A323795.
Cf. A089259, A141268, A271619, A279375, A279785, A306186, A316980, A317533, A318564, A318565, A318566, A330459, A330472.

cp's OEIS Frontend

A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

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A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

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A318563 Number of combinatory separations of strongly normal multisets of weight n.

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A318567 Number of pairs (c, y) where c is an integer composition and y is an integer partition and y can be obtained from c by choosing a partition of each part, flattening, and sorting.

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A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

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A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

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A318816 Regular tetrangle where T(n,k,i) is the number of non-isomorphic multiset partitions of length i of multiset partitions of length k of multisets of size n.

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A330464 Number of non-isomorphic weight-n sets of set-systems with distinct multiset unions.

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