A258466 -id:A258466 - cp's OEIS frontend

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946

Offset: 0

Gus Wiseman, Oct 12 2018

nonn

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022

			The a(9) = 26 multiset partitions:
  {{9}}
  {{1,8}}
  {{2,7}}
  {{3,6}}
  {{4,5}}
  {{1,2,6}}
  {{1,3,5}}
  {{1},{8}}
  {{2,3,4}}
  {{2},{7}}
  {{3,3,3}}
  {{3},{6}}
  {{4},{5}}
  {{1},{2},{6}}
  {{1},{3},{5}}
  {{2},{3},{4}}
  {{3},{3},{3}}
  {{1,1,1,2,2,2}}
  {{1,1,1},{2,2,2}}
  {{1,1,2},{1,2,2}}
  {{1,1},{1,2},{2,2}}
  {{1,2},{1,2},{1,2}}
  {{1,1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,1,1},{1,1,1}}
  {{1},{1},{1},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..150
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.

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]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]

Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022

A320351 Number of connected multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 38, 66, 130, 237, 449, 823, 1538

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 18 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{1,1}}    {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1,2,2}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A319719, A320328, A320330, A320331, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],Length[csm[#]]==1&]],{n,8}]

A262496 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order such that sorts of adjacent parts are different.

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 87, 312, 1269, 5703, 28082, 149643, 855938, 5217753, 33712046, 229799508, 1646314498, 12355371024, 96861186897, 791258791159, 6720627161903, 59234364141343, 540812222291531, 5106663817387466, 49798678281320763, 500857393909589995

Offset: 0

Alois P. Heinz, Sep 24 2015

nonn

			a(3) = 4: 3a, 2a1b, 1a1b1a, 1a1b1c (in this example the sorts are labeled a, b, c).

Alois P. Heinz, Table of n, a(n) for n = 0..400

Row sums of A262495.

Cf. A258466.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
T:= (n, k)->  add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..30);

b[n_, i_, k_] := b[n, i, k] = If[n==0 || i==1, k^(n-1), b[n, i-1, k] + If[i > n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n==0, 1, If[k<2, k, k*b[n, n, k - 1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)

A278644 Number of partitions of n into parts of sorts {1, 2, ... }.

Original entry on oeis.org

1, 1, 4, 17, 95, 649, 5423, 53345, 604570, 7744990, 110596370, 1740967790, 29943077149, 558541778035, 11229820022013, 242071441524480, 5568954194762675, 136181762611151941, 3527284819779421843, 96465042641948254298, 2777679881076121497601

Offset: 0

Alois P. Heinz, Nov 24 2016

nonn

Parts are unordered, sorts are ordered, all sorts up to the highest have to be present.

a(n) mod 2 = A040051(n).

			a(3) = 17: 1a1a1a, 2a1a, 1a, 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a (in this example the sorts are labeled a, b, c).

Alois P. Heinz, Table of n, a(n) for n = 0..424

Row sums of A255970.

Cf. A000670, A008284, A040051, A258466.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

a(n) = Sum_{k=0..n} A255970(n,k).
a(n) = Sum_{k=0..n} A008284(n,k) * A000670(k). - Ludovic Schwob, Sep 25 2023
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 26 2023

A319731 Number of plane partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

Original entry on oeis.org

1, 1, 5, 20, 99, 483, 2855, 16759, 112794, 777862, 5864191, 45388575, 381557427, 3265488790, 29815712658, 279926300139, 2762328453142, 27952237049003, 296275051753578, 3212312177119572, 36258222471852860, 419025393587012853, 5010022284030897550

Offset: 0

Alois P. Heinz, Sep 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Plane partition
Wikipedia, Plane partition

Row sums of A319730.

Cf. A258466.

a(n) = Sum_{k=0..n} A319730(n,k).

A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 87, 138, 211, 324, 486, 727, 1079, 1584, 2305, 3337, 4789, 6830, 9712, 13689, 19225, 26841, 37322, 51598, 71108, 97580, 133350, 181558, 246335, 332991, 448706, 602607, 806732, 1077333, 1433885, 1903682, 2520246, 3328549, 4383929

Offset: 0

Gus Wiseman, May 25 2018

nonn

			The a(6) = 21 unitary factorizations:
(13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)
(2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)
(2*3*5)
The a(6) = 21 multiset partitions:
{{6}}
{{2,4}}
{{1,5}}
{{3,3}}
{{2,2,2}}
{{1,1,4}}
{{1,2,3}}
{{1,1,2,2}}
{{1,1,1,3}}
{{1,1,1,1,2}}
{{1,1,1,1,1,1}}
{{1},{5}}
{{1},{2,3}}
{{2},{4}}
{{2},{1,3}}
{{2},{1,1,1,1}}
{{1,1},{4}}
{{1,1},{2,2}}
{{3},{1,2}}
{{3},{1,1,1}}
{{1},{2},{3}}

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

Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.

Row sums of A321878.

Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz in A321878 *)

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 70, 123, 208, 355, 597

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(6) = 23 antichains:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                  {{1},{2}}  {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1},{3}}    {{1,2,2}}      {{1,1,4}}
                             {{1,1,1,1}}  {{1},{4}}      {{1,2,3}}
                             {{2},{1,1}}  {{2},{3}}      {{1},{5}}
                                          {{1,1,1,2}}    {{2,2,2}}
                                          {{1},{2,2}}    {{2},{4}}
                                          {{3},{1,1}}    {{1,1,1,3}}
                                          {{1,1,1,1,1}}  {{1,1,2,2}}
                                          {{1,1},{1,2}}  {{1},{2,3}}
                                          {{2},{1,1,1}}  {{2},{1,3}}
                                                         {{3},{1,2}}
                                                         {{4},{1,1}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1},{2,2}}
                                                         {{1},{2},{3}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A258466, A261049, A319719, A319721, A320353, A320355, A320356.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,antiQ[#]]&]],{n,10}]

A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 13, 19, 28, 47, 64, 98

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(8) = 19 antichains:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1,3}}    {{1,4}}    {{1,5}}
                {{1},{2}}  {{1},{3}}  {{2,3}}    {{2,4}}
                                      {{1},{4}}  {{1,2,3}}
                                      {{2},{3}}  {{1},{5}}
                                                 {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{2},{1,3}}
                                                 {{3},{1,2}}
                                                 {{1},{2},{3}}
.
  {{7}}          {{8}}
  {{1,6}}        {{1,7}}
  {{2,5}}        {{2,6}}
  {{3,4}}        {{3,5}}
  {{1,2,4}}      {{1,2,5}}
  {{1},{6}}      {{1,3,4}}
  {{2},{5}}      {{1},{7}}
  {{3},{4}}      {{2},{6}}
  {{1},{2,4}}    {{3},{5}}
  {{2},{1,4}}    {{1},{2,5}}
  {{4},{1,2}}    {{1},{3,4}}
  {{1,2},{1,3}}  {{2},{1,5}}
  {{1},{2},{4}}  {{3},{1,4}}
                 {{4},{1,3}}
                 {{5},{1,2}}
                 {{1,2},{1,4}}
                 {{1,2},{2,3}}
                 {{1},{2},{5}}
                 {{1},{3},{4}}

Cf. A001970, A050342, A089259, A258466, A261049, A319719, A319721, A320328, A320331, A320353.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

cp's OEIS Frontend

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

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A320351 Number of connected multiset partitions of integer partitions of n.

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A262496 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order such that sorts of adjacent parts are different.

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Maple

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A278644 Number of partitions of n into parts of sorts {1, 2, ... }.

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Maple

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A319731 Number of plane partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

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A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

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Mathematica

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

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Mathematica

A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

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Mathematica