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%I #17 Jan 02 2025 10:49:57
%S 1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%U 0,0,0,0,1,0,0,0,0,0
%N Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n into parts > 1 with product k.
%C This table counts finite multisets of positive integers > 1 by sum and product. Compare to the triangle A318950.
%F For n <= k we have A(n,k) = A318950(k,n).
%e Array begins:
%e k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
%e -----------------------------------------------
%e n=0: 1 0 0 0 0 0 0 0 0 0 0 0
%e n=1: 0 0 0 0 0 0 0 0 0 0 0 0
%e n=2: 0 1 0 0 0 0 0 0 0 0 0 0
%e n=3: 0 0 1 0 0 0 0 0 0 0 0 0
%e n=4: 0 0 0 2 0 0 0 0 0 0 0 0
%e n=5: 0 0 0 0 1 1 0 0 0 0 0 0
%e n=6: 0 0 0 0 0 1 0 2 1 0 0 0
%e n=7: 0 0 0 0 0 0 1 0 0 1 0 2
%e n=8: 0 0 0 0 0 0 0 1 0 0 0 1
%e n=9: 0 0 0 0 0 0 0 0 1 0 0 0
%e n=10: 0 0 0 0 0 0 0 0 0 1 0 0
%e n=11: 0 0 0 0 0 0 0 0 0 0 1 0
%e n=12: 0 0 0 0 0 0 0 0 0 0 0 1
%e For example, the A(11,48) = 3 partitions are: (4,4,3), (4,3,2,2), (3,2,2,2,2).
%e Antidiagonals begin:
%e n+k=1: 1
%e n+k=2: 0 0
%e n+k=3: 0 0 0
%e n+k=4: 0 0 1 0
%e n+k=5: 0 0 0 0 0
%e n+k=6: 0 0 0 1 0 0
%e n+k=7: 0 0 0 0 0 0 0
%e n+k=8: 0 0 0 0 2 0 0 0
%e n+k=9: 0 0 0 0 0 0 0 0 0
%e n+k=10: 0 0 0 0 0 1 0 0 0 0
%e n+k=11: 0 0 0 0 0 1 0 0 0 0 0
%e n+k=12: 0 0 0 0 0 0 1 0 0 0 0 0
%e n+k=13: 0 0 0 0 0 0 0 0 0 0 0 0 0
%e n+k=14: 0 0 0 0 0 0 2 1 0 0 0 0 0 0
%e n+k=15: 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
%e n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
%e For example, antidiagonal n+k=14 counts the following partitions:
%e n=6: (42), (222)
%e n=7: (7)
%e so the 14th antidiagonal is: (0,0,0,0,0,0,2,1,0,0,0,0,0,0,0).
%t nn=15;
%t tt=Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
%t tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
%t Join@@tr (* sequence *)
%Y Column sums are A001055 = factorizations, strict A045778.
%Y Row sums are A002865 = partitions into parts > 1.
%Y Take transpose and remove upper half (all zeros) to get A318950.
%Y Allowing one gives A379666, antidiagonal sums A379667.
%Y Antidiagonal sums are A379669, zeros A379670.
%Y The strict case allowing ones is A379671, antidiagonal sums A379672.
%Y The strict case is A379678, antidiagonal sums A379679 (zeros A379680).
%Y A000041 counts integer partitions, strict A000009.
%Y A316439 counts factorizations by length, A008284 partitions.
%Y A326622 counts factorizations with integer mean, strict A328966.
%Y Counting and ranking multisets by comparing sum and product:
%Y - same: A001055, ranks A301987
%Y - divisible: A057567, ranks A326155
%Y - divisor: A057568, ranks A326149, see A379733
%Y - greater than: A096276 shifted right, ranks A325038
%Y - greater or equal: A096276, ranks A325044
%Y - less than: A114324, ranks A325037, see A318029
%Y - less or equal: A319005, ranks A379721, see A025147
%Y - different: A379736, ranks A379722, see A111133
%Y Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A325042, A326152, A326178, A379720.
%K nonn,tabl,new
%O 1,33
%A _Gus Wiseman_, Dec 31 2024