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A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.

%I A364912 #13 Dec 13 2024 09:41:51
%S A364912 1,0,1,0,1,2,0,1,2,3,0,1,4,4,5,0,1,4,8,7,7,0,1,6,13,17,12,11,0,1,6,18,
%T A364912 28,30,19,15,0,1,8,24,50,58,53,30,22
%N A364912 Triangle read by rows where T(n,k) is the number of ways to write n as a positive linear combination of an integer partition of k.
%C A364912 A way of writing n as a positive linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i > 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(2,2)) are a way of writing 8 as a positive linear combination of (1,1,2), namely 8 = 3*1 + 1*1 + 2*2.
%H A364912 Steven R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.
%F A364912 As an array, also the number of ways to write n-k as a nonnegative linear combination of an integer partition of k (see programs).
%e A364912 Triangle begins:
%e A364912   1
%e A364912   0  1
%e A364912   0  1  2
%e A364912   0  1  2  3
%e A364912   0  1  4  4  5
%e A364912   0  1  4  8  7  7
%e A364912   0  1  6 13 17 12 11
%e A364912   0  1  6 18 28 30 19 15
%e A364912   0  1  8 24 50 58 53 30 22
%e A364912 Row n = 4 counts the following linear combinations:
%e A364912   .  1*4  2*2      2*1+1*2      4*1
%e A364912           1*1+1*3  1*1+1*1+1*2  3*1+1*1
%e A364912           1*2+1*2  1*1+1*2+1*1  2*1+2*1
%e A364912           1*3+1*1  1*2+1*1+1*1  2*1+1*1+1*1
%e A364912                                 1*1+1*1+1*1+1*1
%e A364912 Row n = 5 counts the following linear combinations:
%e A364912   .  1*5  1*1+1*4  2*1+1*3      3*1+1*2          5*1
%e A364912           1*2+1*3  2*2+1*1      2*1+1*1+1*2      4*1+1*1
%e A364912           1*3+1*2  1*1+1*1+1*3  2*1+1*2+1*1      3*1+2*1
%e A364912           1*4+1*1  1*1+1*2+1*2  1*1+1*1+1*1+1*2  3*1+1*1+1*1
%e A364912                    1*1+1*3+1*1  1*1+1*1+1*2+1*1  2*1+2*1+1*1
%e A364912                    1*2+1*1+1*2  1*1+1*2+1*1+1*1  2*1+1*1+1*1+1*1
%e A364912                    1*2+1*2+1*1  1*2+1*1+1*1+1*1  1*1+1*1+1*1+1*1+1*1
%e A364912                    1*3+1*1+1*1
%e A364912 Array begins:
%e A364912   1   0   0   0    0    0    0     0
%e A364912   1   1   1   1    1    1    1     1
%e A364912   2   2   4   4    6    6    8     8
%e A364912   3   4   8   13   18   24   33    40
%e A364912   5   7   17  28   50   70   107   143
%e A364912   7   12  30  58   108  179  286   428
%e A364912   11  19  53  109  223  394  696   1108
%e A364912   15  30  86  194  420  812  1512  2619
%t A364912 combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A364912 Table[Length[Join@@Table[combp[n,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]
%t A364912 - or -
%t A364912 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A364912 Table[Length[Join@@Table[combs[n-k,ptn],{ptn,IntegerPartitions[k]}]],{n,0,6},{k,0,n}]
%Y A364912 Row k = 0 is A000007.
%Y A364912 Row k = 1 is A000012.
%Y A364912 Column n = 0 is A000041.
%Y A364912 Column n = 1 is A000070.
%Y A364912 Row sums are A006951.
%Y A364912 Row k = 2 is A052928 except initial terms.
%Y A364912 The case of strict integer partitions is A116861.
%Y A364912 Central column is T(2n,n) = A(n,n) = A364907(n).
%Y A364912 With rows reversed we have the nonnegative version A365004.
%Y A364912 A000041 counts integer partitions, strict A000009.
%Y A364912 A008284 counts partitions by length, strict A008289.
%Y A364912 A364350 counts combination-free strict partitions, complement A364839.
%Y A364912 A364913 counts combination-full partitions.
%Y A364912 Cf. A237113, A364272, A364910, A364911, A364914, A364915, A365002.
%K A364912 nonn,tabl
%O A364912 0,6
%A A364912 _Gus Wiseman_, Aug 20 2023