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A240181 Array: t(n,k) is the number of partitions p of n such that the number of distinct numbers in the intersection of p and its conjugate is k, for k >= 0, n >= 1.

%I A240181 #19 Sep 28 2023 18:32:26
%S A240181 0,1,2,2,0,1,2,3,2,4,1,4,6,0,1,4,8,3,8,8,5,1,10,9,11,10,22,8,1,1,14,
%T A240181 22,17,3,18,34,19,5,1,18,50,21,12,26,60,34,13,2,30,74,52,19,0,1,36,
%U A240181 105,57,29,4,44,120,93,34,5,1,60,144,128,40,13,64,186
%N A240181 Array:  t(n,k) is the number of partitions p of n such that the number of distinct numbers in the intersection of p and its conjugate is k, for k >= 0, n >= 1.
%C A240181 First two columns are A240674 and A240675.  Sum of numbers in row n is A000041(n), for n >= 1.  Number of numbers in row n is A240450(n).
%H A240181 Clark Kimberling, <a href="/A240181/b240181.txt">Table of n, a(n) for n = 1..200</a>
%e A240181 First 15 rows:
%e A240181   0 ... 1
%e A240181   2
%e A240181   2 ... 0 ... 1
%e A240181   2 ... 3
%e A240181   2 ... 4 ... 1
%e A240181   4 ... 6 ... 0 ...1
%e A240181   4 ... 8 ... 3
%e A240181   8 ... 8 ... 5 ... 1
%e A240181   10 .. 9 ... 11
%e A240181   10 .. 22 .. 8 ... 1 ... 1
%e A240181   14 .. 22 .. 17 .. 3
%e A240181   18 .. 34 .. 19 .. 5 ... 1
%e A240181   18 .. 50 .. 21 .. 12
%e A240181   26 .. 60 .. 34 .. 13 .. 2
%e A240181   30 .. 74 .. 52 .. 19 .. 0 .. 1
%e A240181 In the following table, p and c(p) denote a partition of 6 and its conjugate:
%e A240181   p ........ c(p)
%e A240181   6 ........ 111111
%e A240181   51 ....... 21111
%e A240181   42 ....... 2211
%e A240181   411 ...... 3111
%e A240181   33 ....... 222
%e A240181   321 ...... 321
%e A240181   3111 ..... 411
%e A240181   222 ...... 33
%e A240181   2211 ..... 42
%e A240181   21111 .... 51
%e A240181   111111 ... 6
%e A240181 Let I(p) be number of numbers in the intersection of c and c(p);  Then I(p) = 0 for 4 choices of p, I(p) = 1 for 6 choices, I(p) = 2 for 0 choices, and I(p) = 3 for 1 choice.  Thus, row 6 is 4 6 0 1.
%t A240181 z = 30; conjugatePartition[part_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[part]; c = Map[BinCounts[#, {0, 1 + Max[#]}] &[Map[Length, Map[Intersection[#, conjugatePartition[#]] &, IntegerPartitions[#]]]] &, Range[z]]; Flatten[c]  (* this sequence *)
%t A240181 Table[Length[c[[n]]], {n, 1, z}] (* A240450 *) (* _Peter J. C. Moses_, Apr 10 2014 *)
%Y A240181 Cf. A240674, A240675, A240450.
%K A240181 nonn,easy,tabf
%O A240181 1,3
%A A240181 _Clark Kimberling_, Apr 12 2014
%E A240181 Name corrected by _Clark Kimberling_, Sep 28 2023