A321655 -id:A321655 - cp's OEIS frontend

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

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).

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}]

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}]

A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 5, 19, 107, 573, 4050, 29093, 249301, 2271020, 23378901, 257871081, 3132494380, 40693204728, 572089068459, 8566311524788, 137165829681775, 2327192535461323, 41865158805428687, 793982154675640340, 15863206077534914434, 332606431999260837036, 7309310804287502958322, 167896287022455809865568

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(3) = 19 matrices:
  [3] [2 1] [1 1 1]
.
  [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Ludovic Schwob, Table of n, a(n) for n = 0..39
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 13.

Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, 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[#]],OrderedQ[Total/@Transpose[prs2mat[#]]]]&]],{n,6}]

Sum of coefficients in the expansions of all homogeneous symmetric functions in terms of monomial symmetric functions. In other words, if Sum_{|y| = n} h(y) = Sum_{|y| = n} c_y * m(y), then a(n) = Sum_{|y| = n} c_y.

a(10) onwards from Ludovic Schwob, Aug 29 2023

A321646 Number of distinct row/column permutations of Ferrers diagrams of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 15, 39, 108, 290, 781, 2050, 5434, 14210, 37150, 96347, 248250, 636278, 1620721, 4108340, 10361338, 26016060, 65019655, 161831393, 401090324, 990229108, 2435316984, 5967684036, 14572351628, 35464928382, 86033632280, 208062026930, 501676936146

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(4) = 15 diagrams:
  o o o o
.
  o o o   o o o   o o o   o o   o         o         o
  o         o         o   o o   o o o   o o o   o o o
.
  o o   o o   o     o       o     o
  o       o   o o   o     o o     o
  o       o   o     o o     o   o o
.
  o
  o
  o
  o

Cf. A000219, A008480, A049311, A068313, A101370, A120733, A122111, A321645, A321647, A321648, A321655.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Sum[Length[Permutations[y]]*Length[Permutations[conj[y]]],{y,IntegerPartitions[n]}],{n,10}]

a(n) = Sum_{k = 1..A000041(n)} A008480(A215366(n,k)) * A008480(A122111(A215366(n,k))).

a(11)-a(30) from Alois P. Heinz, Nov 15 2018

A321647 Number of distinct row/column permutations of the Ferrers diagram of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 6, 1, 8, 6, 1, 1, 6, 1, 9, 12, 10, 1, 8, 1, 12, 1, 12, 1, 36, 1, 1, 20, 14, 8, 12, 1, 16, 30, 12, 1, 72, 1, 15, 9, 18, 1, 10, 1, 9, 42, 18, 1, 8, 20, 16, 56, 20, 1, 72, 1, 22, 18, 1, 40, 120, 1, 21, 72, 72, 1, 20, 1, 24, 9, 24, 10, 180, 1, 15, 1, 26, 1, 144, 70, 28, 90, 20, 1, 72, 30, 27, 110, 30, 112, 12, 1, 12

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(10) = 6 permutations:
  o o   o o   o     o       o     o
  o       o   o o   o     o o     o
  o       o   o     o o     o   o o
The a(21) = 12 permutations:
  o o   o o   o o   o o   o o   o o   o     o     o       o     o     o
  o o   o o   o     o       o     o   o o   o o   o     o o   o o     o
  o       o   o o   o     o o     o   o o   o     o o   o o     o   o o
  o       o   o     o o     o   o o   o     o o   o o     o   o o   o o

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000219, A008480, A049311, A056239, A068313, A101370, A112798, A122111, A296150, A321645, A321646, A321648, A321655.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Permutations[primeMS[n]]]*Length[Permutations[conj[primeMS[n]]]],{n,50}]

A008480(n) = {my(sig=factor(n)[, 2]); vecsum(sig)!/factorback(apply(k->k!, sig))}; \\ From A008480
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A321647(n) = (A008480(n) * A008480(A122111(n))); \\ Antti Karttunen, Feb 09 2019

a(n) = A008480(n) * A008480(A122111(n)) = A008480(n) * A321648(n).

More terms from Antti Karttunen, Feb 09 2019

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.

Original entry on oeis.org

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

cp's OEIS Frontend

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

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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.

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A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

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A321646 Number of distinct row/column permutations of Ferrers diagrams of integer partitions of n.

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A321647 Number of distinct row/column permutations of the Ferrers diagram of the integer partition with Heinz number n.

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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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