A321662 -id:A321662 - cp's OEIS frontend

A321659 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose nonzero entries are all distinct.

1, 1, 1, 9, 9, 17, 161, 169, 313, 465, 5313, 5465, 10457, 15313, 25009, 271929, 286329, 537953, 799121, 1297369, 1805161, 20532897, 21292017, 40508297, 59738825, 97431073, 135137569, 209525865, 2089381929, 2200470833, 4135252289, 6124698121, 9937836505

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(5) = 17 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [4 0] [3] [3 0] [2] [2 0] [1] [1 0] [0 4] [0 3] [0 2] [0 1]
  [1] [0 1] [2] [0 2] [3] [0 3] [4] [0 4] [1 0] [2 0] [3 0] [4 0]

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

Cf. A000219, A007716, A008289, A101370, A114736, A117433, A120733, A321645, A321653, A321655, A321660, A321661, A321662.

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]]&]],{n,5}]

\\ here b(n) is A101370(n).
b(n)={sum(m=0, n, sum(k=0, m, stirling(m,k,2)*k!)^2*polcoef(log(1+x+O(x*x^n))^m, n)/m!)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*i!*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} A101370(k)*k!*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 45, 49, 85, 125, 233, 273, 417, 529, 745, 2573, 2861, 4761, 6837, 10489, 14317, 22637, 28289, 40041, 52041, 70177, 88561, 117605, 234773, 274761, 407469, 553681, 792613, 1052525, 1493033, 1959009, 3135537, 3904129, 5475673, 7173725, 9853325

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(5) = 9 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [3] [2] [1]
  [1] [2] [3] [4]

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

Cf. A000005, A000219, A007716, A008289, A114736, A117433, A120733, A321645, A321653, A321655, A321659, A321661, A321662.

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@@Join@@prs2mat[#]]&]],{n,5}]

seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, (n-1)!*numdiv(n-1) + n!*(numdiv(n) - 2)))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} (k!*A000005(k) + (k+1)!*(A000005(k+1) - 2))*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A321661 Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 22, 25, 40, 58, 186, 204, 347, 478, 734, 2033, 2402, 3814, 5464, 8142, 11058, 30142, 34437, 55940, 77794, 116954, 156465, 229462, 533612, 640544, 994922, 1397896, 2048316, 2778750, 3987432, 5292293, 11921070, 14076550, 21802928, 29917842, 44080285

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, whose nonzero entries are all distinct.
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(6) = 22 multiset partitions:
  {{1}}  {{11}}  {{111}}    {{1111}}    {{11111}}    {{111111}}
                 {{122}}    {{1222}}    {{11222}}    {{112222}}
                 {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}
                 {{1}{22}}  {{1}{222}}  {{1}{1111}}  {{122333}}
                                        {{11}{111}}  {{1}{11111}}
                                        {{11}{222}}  {{11}{1111}}
                                        {{1}{2222}}  {{1}{11222}}
                                                     {{11}{1222}}
                                                     {{11}{2222}}
                                                     {{112}{222}}
                                                     {{11}{2333}}
                                                     {{1}{22222}}
                                                     {{122}{222}}
                                                     {{1}{22333}}
                                                     {{122}{333}}
                                                     {{2}{11222}}
                                                     {{22}{1222}}
                                                     {{1}{11}{111}}
                                                     {{1}{11}{222}}
                                                     {{1}{22}{222}}
                                                     {{1}{22}{333}}
                                                     {{2}{11}{222}}

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

Cf. A000219, A007716, A008289, A059201, A059849, A114736, A117433, A120733, A321653, A321659, A321660, A321662.

\\ here b(n) is A059849(n).
b(n)={sum(k=0, n, stirling(n,k,1)*sum(i=0, k, stirling(k,i,2))^2)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p,i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} A059849(k)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A323657 Number of strict solid partitions of n.

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 16, 19, 28, 40, 82, 94, 145, 190, 274, 463, 580, 802, 1096, 1486, 1948, 3148, 3811, 5314, 6922, 9394, 11971, 16156, 23044, 28966, 38368, 50002, 65116, 83872, 108706, 137917, 192070, 236242, 308698, 390772, 506935, 633982, 817324, 1018090

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

A strict solid partition is an infinite three-dimensional array of distinct positive integers (and any number of zeros) summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros.

			The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))       ((5))       ((6))
                ((21))      ((31))      ((32))      ((42))
                ((2)(1))    ((3)(1))    ((41))      ((51))
                ((2))((1))  ((3))((1))  ((3)(2))    ((321))
                                        ((4)(1))    ((4)(2))
                                        ((3))((2))  ((5)(1))
                                        ((4))((1))  ((31)(2))
                                                    ((32)(1))
                                                    ((4))((2))
                                                    ((5))((1))
                                                    ((31))((2))
                                                    ((3)(2)(1))
                                                    ((32))((1))
                                                    ((3)(1))((2))
                                                    ((3)(2))((1))
                                                    ((3))((2))((1))

Alois P. Heinz, Table of n, a(n) for n = 0..740 (first 401 terms from John Tyler Rascoe)

Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A008289, A114736, A117433 (strict plane partitions), A207542, A321662, A323657.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}]
Table[Length[Join@@Table[Select[Tuples[strplptns/@y],And[UnsameQ@@Flatten[#],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])]&],{y,IntegerPartitions[n]}]],{n,10}]

a(n) = Sum_{k=1..n} A008289(n,k)*A207542(k) for n > 0. - John Tyler Rascoe, Dec 19 2024

a(21) onwards from John Tyler Rascoe, Dec 19 2024

A002974 Number of restricted solid partitions of n.

Original entry on oeis.org

1, 1, 4, 7, 11, 20, 35, 59, 99, 165, 270, 443, 723, 1161, 1861, 2961, 4654, 7279, 11317, 17476, 26879, 41132, 62601, 94878, 143172, 215115, 321995, 480216, 713655, 1057192

Offset: 1

N. J. A. Sloane

Definition, based on Math. Review MR0297583: By a solid partition of n is meant a 3-dimensional arrangement of positive integers N(x,y,z) satisfying the conditions (i) the integer N(x,y,z) is located at the point with Cartesian coordinates (x,y,z); N(x,y,z) is defined only for certain integers x,y,z >= 0, and (ii) if N(x,y,z) is defined and 0 <= x' <= x, 0 <= y' <= y, 0 <= z' <= z then N(x,y,z) is defined and N(x',y',z') <= N(x,y,z). A solid partition is said to correspond to an (ordinary) partition of n=n_1+n_2+...+n_t, n_k>0, if there is a one-to-one correspondence between the summands n_k and the points (x_k,y_k,z_k) for which N is defined so that n_k=N(x_k,y_k,z_k). Finally, a restricted solid partition is a solid partition such that x'<=x, y'<=y, z'<=z and N(x',y',z')=N(x,y,z) implies x'=x, y'=y, z'=z.

Alternatively, a restricted solid partition is an infinite three-dimensional array of nonnegative integers summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros. - Gus Wiseman, Jan 22 2019

			From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(6) = 20 restricted solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))          ((5))           ((6))
                ((21))      ((31))         ((32))          ((42))
                ((2)(1))    ((3)(1))       ((41))          ((51))
                ((2))((1))  ((21)(1))      ((3)(2))        ((321))
                            ((3))((1))     ((4)(1))        ((4)(2))
                            ((21))((1))    ((31)(1))       ((5)(1))
                            ((2)(1))((1))  ((3))((2))      ((31)(2))
                                           ((4))((1))      ((32)(1))
                                           ((31))((1))     ((41)(1))
                                           ((3)(1))((1))   ((4))((2))
                                           ((21)(1))((1))  ((5))((1))
                                                           ((31))((2))
                                                           ((3)(2)(1))
                                                           ((32))((1))
                                                           ((41))((1))
                                                           ((3)(1))((2))
                                                           ((3)(2))((1))
                                                           ((4)(1))((1))
                                                           ((31)(1))((1))
                                                           ((3))((2))((1))
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Gupta, Restricted solid partitions, J. Combin. Theory, A 13 (1972), 140-144.

Cf. A000219, A000293 (solid partitions), A000334, A001970, A114736 (restricted plane partitions), A117433 (strict plane partitions), A321662, A323657 (strict solid partitions).

srcplptns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And[And@@(GreaterEqual@@@Transpose[PadRight[#]]),And@@Greater@@@#,And@@(Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}])]&],{ptn,IntegerPartitions[n]}];
srcsolids[n_]:=Join@@Table[Select[Tuples[srcplptns/@y],And[And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)]),And@@(Greater@@@DeleteCases[Transpose[Join@@@(PadRight[#,{n,n}]&/@#)],0,{2}])]&],{y,IntegerPartitions[n]}]
Table[Length[srcsolids[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from Sean A. Irvine, Dec 15 2014

cp's OEIS Frontend

A321659 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose nonzero entries are all distinct.

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A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

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A321661 Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.

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A323657 Number of strict solid partitions of n.

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A002974 Number of restricted solid partitions of n.

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