This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379671 #7 Jan 02 2025 21:27:51
%S A379671 1,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,
%T A379671 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,
%U A379671 0,0,0,0,2,0,0,0,0,0
%N A379671 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers with sum n and product k.
%C A379671 Counts finite sets of positive integers by sum and product.
%e A379671 Array begins:
%e A379671 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
%e A379671 -----------------------------------------------
%e A379671 n=0: 1 0 0 0 0 0 0 0 0 0 0 0
%e A379671 n=1: 1 0 0 0 0 0 0 0 0 0 0 0
%e A379671 n=2: 0 1 0 0 0 0 0 0 0 0 0 0
%e A379671 n=3: 0 1 1 0 0 0 0 0 0 0 0 0
%e A379671 n=4: 0 0 1 1 0 0 0 0 0 0 0 0
%e A379671 n=5: 0 0 0 1 1 1 0 0 0 0 0 0
%e A379671 n=6: 0 0 0 0 1 2 0 1 0 0 0 0
%e A379671 n=7: 0 0 0 0 0 1 1 1 0 1 0 1
%e A379671 n=8: 0 0 0 0 0 0 1 1 0 1 0 2
%e A379671 n=9: 0 0 0 0 0 0 0 1 1 0 0 1
%e A379671 n=10: 0 0 0 0 0 0 0 0 1 1 0 0
%e A379671 n=11: 0 0 0 0 0 0 0 0 0 1 1 0
%e A379671 n=12: 0 0 0 0 0 0 0 0 0 0 1 1
%e A379671 The A(8,12) = 2 sets are: {2,6}, {1,3,4}.
%e A379671 The A(14,40) = 2 sets are: {4,10}, {1,5,8}.
%e A379671 Antidiagonals begin:
%e A379671 n+k=1: 1
%e A379671 n+k=2: 0 1
%e A379671 n+k=3: 0 0 0
%e A379671 n+k=4: 0 0 1 0
%e A379671 n+k=5: 0 0 0 1 0
%e A379671 n+k=6: 0 0 0 1 0 0
%e A379671 n+k=7: 0 0 0 0 1 0 0
%e A379671 n+k=8: 0 0 0 0 1 0 0 0
%e A379671 n+k=9: 0 0 0 0 0 1 0 0 0
%e A379671 n+k=10: 0 0 0 0 0 1 0 0 0 0
%e A379671 n+k=11: 0 0 0 0 0 1 1 0 0 0 0
%e A379671 n+k=12: 0 0 0 0 0 0 2 0 0 0 0 0
%e A379671 n+k=13: 0 0 0 0 0 0 0 1 0 0 0 0 0
%e A379671 n+k=14: 0 0 0 0 0 0 1 1 0 0 0 0 0 0
%e A379671 n+k=15: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
%e A379671 n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
%e A379671 For example, antidiagonal n+k=11 counts the following sets:
%e A379671 n=5: {2,3}
%e A379671 n=6: {1,5}
%e A379671 so the 11th antidiagonal is: (0,0,0,0,0,1,1,0,0,0,0).
%t A379671 nn=12;
%t A379671 tt=Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
%t A379671 tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
%t A379671 Join@@tr (* sequence *)
%Y A379671 Row sums are A000009 = strict partitions, non-strict A000041.
%Y A379671 Column sums are 2*A045778 where A045778 = strict factorizations, non-strict A001055.
%Y A379671 Antidiagonal sums are A379672, non-strict A379667 (zeros A379670).
%Y A379671 Without ones we have A379678, antidiagonal sums A379679 (zeros A379680).
%Y A379671 The non-strict version is A379666, without ones A379668.
%Y A379671 A316439 counts factorizations by length, partitions A008284.
%Y A379671 A326622 counts factorizations with integer mean, strict A328966.
%Y A379671 Counting and ranking multisets by comparing sum and product:
%Y A379671 - same: A001055, ranks A301987
%Y A379671 - divisible: A057567, ranks A326155
%Y A379671 - divisor: A057568, ranks A326149, see A379733
%Y A379671 - greater than: A096276 shifted right, ranks A325038
%Y A379671 - greater or equal: A096276, ranks A325044
%Y A379671 - less than: A114324, ranks A325037, see A318029
%Y A379671 - less or equal: A319005, ranks A379721, see A025147
%Y A379671 - different: A379736, ranks A379722, see A111133
%Y A379671 Cf. A002865, A003963, A028422, A069016, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.
%K A379671 nonn,tabl
%O A379671 1,73
%A A379671 _Gus Wiseman_, Jan 01 2025