A101509 -id:A101509 - cp's OEIS frontend

A306320 Number of square plane partitions of n with distinct row sums and distinct column sums.

1, 1, 1, 1, 1, 2, 2, 5, 5, 10, 11, 18, 21, 31, 37, 56, 70, 97, 134, 180, 247, 343, 462, 623, 850, 1128, 1509, 2004, 2649, 3467, 4590, 5958, 7814, 10161, 13287, 17208, 22495, 29129, 37997, 49229, 64098, 82940, 107868, 139390, 180737, 233214, 301527, 388018, 500058

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

			The a(12) = 21 square plane partitions with distinct row sums and distinct column sums:
[twelve]
.
[64][73][82][91][54][63][72][81][44][53][53][62][62][71][43][43][52][52][61]
[11][11][11][11][21][21][21][21][31][22][31][22][31][31][32][41][32][41][41]
.
[221]
[211]
[111]

Cf. A000219, A089299 (square plane partitions), A101509, A271619, A279785, A306318, A323429, A323529, A323530, A323531.

Table[Sum[Length[Select[Union[Reverse/@Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Total/@#&&UnsameQ@@Total/@If[#=={},{},Transpose[#]]&&And@@OrderedQ/@Reverse/@If[#=={},{},Transpose[#]]&]],{ptn,IntegerPartitions[n]}],{n,0,20}]

A159297 Number of 3D matrices with positive integer entries such that sum of all entries equals n.

Original entry on oeis.org

1, 4, 10, 25, 58, 130, 286, 620, 1329, 2827, 5977, 12559, 26227, 54493, 112849, 233272, 481616, 992955, 2043238, 4194649, 8591014, 17559133, 35833948, 73054885, 148849186, 303171755, 617306563, 1256452642, 2555937826

Offset: 1

Lior Manor, Apr 09 2009

nonn

Equivalently, number of quadruples (i, j, k; P) such that i, j and k are positive integers and P is a composition of n into ijk parts. (A composition of n with m parts is an ordered list of m positive integers that sum to n. The number of compositions of n into m parts is given by the binomial coefficient C(n - 1, m - 1).) [Joel B. Lewis, May 07 2009]

			For n=3, the 10 possible matrices are: 3 (1*1*1); (1,2) as three different vectors (1*1*2, 1*2*1, 2*1*1); (2,1) as three different vectors (1*1*2, 1*2*1, 2*1*1); and (1,1,1) as three different vectors (1*1*3, 1*3*1, 3*1*1). [Typo corrected by _Joel B. Lewis_, Apr 04 2011]

Cf. A101509

Table[Sum[Sum[Sum[Binomial[n - 1, i*j*k - 1], {i, 1, n}], {j, 1, n}], {k, 1, n}], {n, 1, 40}] (* Joel B. Lewis, May 07 2009 *)

a(n) = sum(C(n - 1, ijk - 1)) where the sum is over all triples (i, j, k) such that 0 < i, j, k and ijk <= n. [Joel B. Lewis, May 07 2009]

More terms from Joel B. Lewis, May 07 2009

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A306320 Number of square plane partitions of n with distinct row sums and distinct column sums.

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