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 A379666 #7 Jan 02 2025 21:27:58
%S A379666 1,0,1,0,0,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,2,1,
%T A379666 1,1,0,0,0,0,0,2,1,1,1,0,0,0,0,0,1,2,1,1,1,0,0,0,0,0,1,1,2,1,1,1,0,0,
%U A379666 0,0,0,0,2,1,2,1,1,1
%N A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.
%C A379666 Counts finite multisets of positive integers by sum and product.
%e A379666 Array begins:
%e A379666 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
%e A379666 -----------------------------------------------
%e A379666 n=0: 1 0 0 0 0 0 0 0 0 0 0 0
%e A379666 n=1: 1 0 0 0 0 0 0 0 0 0 0 0
%e A379666 n=2: 1 1 0 0 0 0 0 0 0 0 0 0
%e A379666 n=3: 1 1 1 0 0 0 0 0 0 0 0 0
%e A379666 n=4: 1 1 1 2 0 0 0 0 0 0 0 0
%e A379666 n=5: 1 1 1 2 1 1 0 0 0 0 0 0
%e A379666 n=6: 1 1 1 2 1 2 0 2 1 0 0 0
%e A379666 n=7: 1 1 1 2 1 2 1 2 1 1 0 2
%e A379666 n=8: 1 1 1 2 1 2 1 3 1 1 0 3
%e A379666 n=9: 1 1 1 2 1 2 1 3 2 1 0 3
%e A379666 n=10: 1 1 1 2 1 2 1 3 2 2 0 3
%e A379666 n=11: 1 1 1 2 1 2 1 3 2 2 1 3
%e A379666 n=12: 1 1 1 2 1 2 1 3 2 2 1 4
%e A379666 For example, the A(9,12) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
%e A379666 Antidiagonals begin:
%e A379666 n+k=1: 1
%e A379666 n+k=2: 0 1
%e A379666 n+k=3: 0 0 1
%e A379666 n+k=4: 0 0 1 1
%e A379666 n+k=5: 0 0 0 1 1
%e A379666 n+k=6: 0 0 0 1 1 1
%e A379666 n+k=7: 0 0 0 0 1 1 1
%e A379666 n+k=8: 0 0 0 0 2 1 1 1
%e A379666 n+k=9: 0 0 0 0 0 2 1 1 1
%e A379666 n+k=10: 0 0 0 0 0 1 2 1 1 1
%e A379666 n+k=11: 0 0 0 0 0 1 1 2 1 1 1
%e A379666 n+k=12: 0 0 0 0 0 0 2 1 2 1 1 1
%e A379666 n+k=13: 0 0 0 0 0 0 0 2 1 2 1 1 1
%e A379666 n+k=14: 0 0 0 0 0 0 2 1 2 1 2 1 1 1
%e A379666 n+k=15: 0 0 0 0 0 0 1 2 1 2 1 2 1 1 1
%e A379666 n+k=16: 0 0 0 0 0 0 0 1 3 1 2 1 2 1 1 1
%e A379666 For example, antidiagonal n+k=10 counts the following partitions:
%e A379666 n=5: (5)
%e A379666 n=6: (411), (2211)
%e A379666 n=7: (31111)
%e A379666 n=8: (2111111)
%e A379666 n=9: (111111111)
%e A379666 so the 10th antidiagonal is: (0,0,0,0,0,1,2,1,1,1).
%t A379666 nn=12;
%t A379666 tt=Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
%t A379666 tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
%t A379666 Join@@tr (* sequence *)
%Y A379666 Row sums are A000041 = partitions of n, strict A000009, no ones A002865.
%Y A379666 Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
%Y A379666 Antidiagonal sums are A379667.
%Y A379666 The case without ones is A379668, antidiagonal sums A379669 (zeros A379670).
%Y A379666 The strict case is A379671, antidiagonal sums A379672.
%Y A379666 The strict case without ones is A379678, antidiagonal sums A379679 (zeros A379680).
%Y A379666 A316439 counts factorizations by length, partitions A008284.
%Y A379666 A326622 counts factorizations with integer mean, strict A328966.
%Y A379666 Counting and ranking multisets by comparing sum and product:
%Y A379666 - same: A001055, ranks A301987
%Y A379666 - divisible: A057567, ranks A326155
%Y A379666 - divisor: A057568, ranks A326149, see A379733
%Y A379666 - greater than: A096276 shifted right, ranks A325038
%Y A379666 - greater or equal: A096276, ranks A325044
%Y A379666 - less than: A114324, ranks A325037, see A318029
%Y A379666 - less or equal: A319005, ranks A379721, see A025147
%Y A379666 - different: A379736, ranks A379722, see A111133
%Y A379666 Cf. A003963, A028422, A069016, A096765, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.
%K A379666 nonn,tabl
%O A379666 1,33
%A A379666 _Gus Wiseman_, Jan 01 2025