A319646 -id:A319646 - cp's OEIS frontend

A319616 Number of non-isomorphic square multiset partitions of weight n.

1, 1, 2, 4, 11, 27, 80, 230, 719, 2271, 7519, 25425, 88868, 317972, 1168360, 4392724, 16903393, 66463148, 266897917, 1093550522, 4568688612, 19448642187, 84308851083, 371950915996, 1669146381915, 7615141902820, 35304535554923, 166248356878549, 794832704948402, 3856672543264073, 18984761300310500

Offset: 0

Gus Wiseman, Sep 25 2018

nonn

A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices.

Also the number of square integer matrices with entries summing to n and no empty rows or columns, up to permutation of rows and columns.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1}, {2}}
3: {{1,1,1}}
   {{1}, {2,2}}
   {{2}, {1,2}}
   {{1}, {2},{3}}
4: {{1,1,1,1}}
   {{1}, {1,2,2}}
   {{1}, {2,2,2}}
   {{2}, {1,2,2}}
   {{1,1}, {2,2}}
   {{1,2}, {1,2}}
   {{1,2}, {2,2}}
   {{1}, {1}, {2,3}}
   {{1}, {2}, {3,3}}
   {{1}, {3}, {2,3}}
   {{1}, {2}, {3}, {4}}
Non-isomorphic representatives of the a(4) = 11 square matrices:
. [4]
.
. [1 0]   [1 0]   [0 1]   [2 0]   [1 1]   [1 1]
. [1 2]   [0 3]   [1 2]   [0 2]   [1 1]   [0 2]
.
. [1 0 0]   [1 0 0]   [1 0 0]
. [1 0 0]   [0 1 0]   [0 0 1]
. [0 1 1]   [0 0 2]   [0 1 1]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A321615.

Cf. A000219, A007716, A007718, A056156, A059201, A316980, A316983, A318795, A319560, A319616-A319646, A300913.

(* See A318795 for M[m, n, k]. *)
T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];
a[0] = 1; a[n_] := Sum[T[n, k], {k, 1, n}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 16}] (* Jean-François Alcover, Nov 24 2018, after Andrew Howroyd *)

\\ See A318795 for M.
a(n) = {if(n==0, 1, sum(i=1, n, M(i,i,n) - 2*M(i,i-1,n) + M(i-1,i-1,n)))} \\ Andrew Howroyd, Nov 15 2018

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n,polcoef(G(k,n,n,y),k,y) - polcoef(G(k-1,n,n,y),k,y)))} \\ Andrew Howroyd, Jan 15 2024

a(11)-a(20) from Andrew Howroyd, Nov 15 2018

a(21) onwards from Andrew Howroyd, Jan 15 2024

A319721 Number of non-isomorphic antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 4, 8, 24, 50, 148, 349, 1014, 2717, 8114

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{1},{1}}
   {{1},{2},{2}}
   {{1},{2},{3}}

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

Cf. A001055, A001970, A007716, A096827, A253249, A285572, A285573, A293993, A318099, A319616-A319646, A319719.

A318099 Number of non-isomorphic weight-n antichains of (not necessarily distinct) multisets whose dual is also an antichain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 4, 7, 19, 32, 81, 142, 337, 659, 1564

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 7 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{1},{1}}
   {{1},{2},{2}}
   {{1},{2},{3}}

Cf. A000219, A006126, A007716, A049311, A059201, A283877, A306007, A316980, A316983, A319558, A319560, A319616-A319646, A300913.

A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

Original entry on oeis.org

1, 1, 3, 29, 1885, 18658259

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated elements).

			Non-isomorphic representatives of the a(3) = 29 covers:
   {{1,3},{2,3}}
   {{1},{2},{3}}
   {{1},{3},{2,3}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{3},{1,3},{2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2,3}}
   {{1},{2},{1,3},{2,3}}
   {{2},{3},{1,3},{2,3}}
   {{1},{3},{2,3},{1,2,3}}
   {{2},{3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3}}
   {{2},{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,3},{2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{3},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A316980, A316983, A318099, A319558, A319559, A319616-A319646, A300913.

a(5) from Max Alekseyev, Jul 13 2022

A114736 Number of planar partitions of n where parts strictly decrease along each row and column.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 10, 15, 22, 33, 49, 70, 102, 146, 205, 290, 405, 561, 779, 1071, 1463, 1999, 2714, 3667, 4946, 6641, 8880, 11848, 15753, 20870, 27586, 36354, 47766, 62621, 81878, 106785, 138975, 180449, 233778, 302270, 390027, 502256, 645603, 828330, 1060851

Offset: 0

Franklin T. Adams-Watters, Mar 16 2006

nonn

If these partitions are "flattened" into a simple partition, the resulting partitions are those for which any part size present with multiplicity k implies the presence of at least k(k-1)/2 larger parts. E.g., [3,1|1] flattens to [3,1^2], 1 has multiplicity 2, so there must be at least 2*1/2 = 1 part larger than 1 - which is the 3.

			For n = 5, we have the 6 partitions [5], [4,1], [4|1], [3,2], [3|2] and [3,1|1].
From _Gus Wiseman_, Nov 15 2018: (Start)
The a(6) = 10 plane partitions:
  6   5 1   4 2   3 2 1
.
  5   4 1   4   3 2   3 1
  1   1     2   1     2
.
  3
  2
  1
(End)

B. Gordon, Multirowed partitions with strict decrease along columns (Notes on plane partitions IV.), Symposia Amer. Math. Soc. 19 (1971) 91-100.

Alois P. Heinz, Table of n, a(n) for n = 0..85

Cf. A000009, A000219, A001970, A007716, A068313, A117433, A120733, A319646, A321645, A321652, A321653, A321655.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],And@@(OrderedQ[#,Greater]&/@prs2mat[#]),And@@(OrderedQ[#,Greater]&/@Transpose[prs2mat[#]])]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)

Clarified definition, added 30 terms and reference. - Dennis K Moore, Jan 12 2011

a(40)-a(44) from Alois P. Heinz, Sep 26 2018

A117433 Number of planar partitions of n with all part sizes distinct.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 149, 187, 243, 321, 413, 527, 735, 895, 1165, 1467, 1885, 2335, 2997, 3853, 4765, 5977, 7473, 9269, 11531, 14255, 17537, 22201, 26897, 33233, 40613, 50027, 60637, 74459, 89963, 109751, 134407, 162117, 195859

Offset: 0

Franklin T. Adams-Watters, Mar 16 2006, Apr 01 2008

nonn

Matches A072706 for n < 10, since a unimodal composition into distinct parts can be placed uniquely as a hook. Starting with n = 10, additional partitions are possible (starting with [4,3|2,1] and [4,2|3,1]).

			From _Gus Wiseman_, Nov 15 2018: (Start)
The a(10) = 35 strict plane partitions (A = 10):
  A  64  73  82  532  91  541  631  721  4321
.
  9  54  63  72  432  8  53  71  431  7  43  52  61  421  6  42  51
  1  1   1   1   1    2  2   2   2    3  21  3   3   3    4  31  4
.
  7  6  5  43  42  5  41
  2  3  4  2   3   3  3
  1  1  1  1   1   2  2
.
  4
  3
  2
  1
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Franklin T. Adams-Watters)
OEIS Wiki, Plane partitions

Cf. A000219, A072706, A117434, A000009.

Cf. A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321652, A321653, A321655, A321659, A321660.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
g:= proc(n) g(n):= `if`(n<2, 1, (n-1)*g(n-2) +g(n-1)) end:
a:= proc(n) b(n, n); add(%[i]*g(i-1), i=1..nops(%)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 18 2012

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@DeleteCases[Join@@prs2mat[#],0],And@@(OrderedQ[#,Greater]&/@prs2mat[#]),And@@(OrderedQ[#,Greater]&/@Transpose[prs2mat[#]])]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]];
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, zip[Plus, b[n, i - 1], If[i > n, {}, Join[{0}, b[n - i, i - 1]]], 0]]];
g[n_] := g[n] = If[n < 2, 1, (n - 1)*g[n - 2] + g[n - 1]];
a[n_] := With[{bn = b[n, n]}, Sum[bn[[i]]*g[i - 1], {i, 1, Length[bn]}]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} A000085(k)*A008289(n,k).

A319719 Number of non-isomorphic connected antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 3, 4, 10, 14, 48, 95, 305, 822, 2615

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices. Connected antichains are also called clutters.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 10 connected antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{1},{1},{1}}

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

Cf. A001055, A001970, A007716, A007718, A056156, A096827, A253249, A285573, A293994, A318099, A319557, A319616-A319646, A319721.

A321645 Number of distinct row/column permutations of plane partitions of n.

Original entry on oeis.org

1, 1, 3, 11, 32, 96, 290, 864, 2502, 7134, 20081

Offset: 0

Gus Wiseman, Nov 15 2018

			The a(3) = 11 permutations of plane partitions:
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1] [1 1] [1] [1 0] [0 1]
  [1] [1 0] [0 1] [2] [1 1] [1 1]
.
  [1]
  [1]
  [1]

Cf. A000219, A001970, A007716, A008480, A068313, A114736, A120733, A316983, A319646.

Cf. A321646, A321647, A321648, A321652, A321653, A321655.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Sort[Map[Last,GatherBy[Sort[Reverse/@#],First],{2}],submultisetQ],submultisetQ],OrderedQ[Sort[Sort/@Map[Last,GatherBy[#,First],{2}],submultisetQ],submultisetQ]]&]],{n,6}]

A319639 Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 1, 2, 20, 2043

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

			The a(1) = 1 through a(3) = 2 antichain covers:
1: {{1}}
2: {{1},{2}}
3: {{1},{2},{3}}
   {{1,2},{1,3},{2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

A321653 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with strictly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 14, 44, 72, 147, 381, 1405

Offset: 0

Gus Wiseman, Nov 15 2018

			The a(5) = 14 matrices:
  [5] [4 1] [3 2]
.
  [4] [4 0] [3 1] [3 1] [3] [3 0] [3 0] [2 2] [2 1] [2 1] [1 2]
  [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1] [2 0]

Cf. A000219, A001970, A007716, A068313, A114736, A117433, A120733, A319646, A321645, A321652, A321654, A321655.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],OrderedQ[Total/@prs2mat[#],Greater],OrderedQ[Total/@Transpose[prs2mat[#]],Greater]]&]],{n,6}]

cp's OEIS Frontend

A319616 Number of non-isomorphic square multiset partitions of weight n.

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A319721 Number of non-isomorphic antichains of multisets of weight n.

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A318099 Number of non-isomorphic weight-n antichains of (not necessarily distinct) multisets whose dual is also an antichain of (not necessarily distinct) multisets.

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A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

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A114736 Number of planar partitions of n where parts strictly decrease along each row and column.

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A117433 Number of planar partitions of n with all part sizes distinct.

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A319719 Number of non-isomorphic connected antichains of multisets of weight n.

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A321645 Number of distinct row/column permutations of plane partitions of n.

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A319639 Number of antichain covers of n vertices by distinct sets whose dual is also an antichain of distinct sets.

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