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A230025 Triangular array: t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n.

%I A230025 #21 Oct 24 2023 10:33:20
%S A230025 1,0,2,1,0,2,1,2,0,2,1,2,2,0,2,1,2,4,2,0,2,1,4,2,4,2,0,2,2,2,6,2,6,2,
%T A230025 0,2,2,6,2,8,2,6,2,0,2,2,4,12,2,8,2,8,2,0,2,2,8,6,14,2,10,2,8,2,0,2,3,
%U A230025 6,14,8,18,2,10,2,10,2,0,2,3,10,10,20,10
%N A230025 Triangular array:  t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n.
%C A230025 Definitions: the self-conjugate portion of a partition p is the portion of the Ferrers graph of p that remains unchanged when p is reflected about its principal diagonal.  The outliers of p are the nodes of the Ferrers graph that lie outside the self-conjugate portion of p.
%C A230025 Sum of numbers in row n is A000041(n).
%C A230025 t(n,k) is the number of partitions p of n such that d(p,p*) = k, where d is the distance function introduced in A366156 and p* is the conjugate of p. - _Clark Kimberling_, Oct 03 2023
%e A230025 The first 9 rows:
%e A230025   1
%e A230025   0 2
%e A230025   1 0 2
%e A230025   1 2 0 2
%e A230025   1 2 2 0 2
%e A230025   1 2 4 2 0 2
%e A230025   1 4 2 4 2 0 2
%e A230025   2 2 6 2 6 2 0 2
%e A230025   2 6 2 8 2 6 2 0 2
%e A230025 The Ferrers graph of the partition p = [4,4,1,1] of 10 follows:
%e A230025   1 1 1 1
%e A230025   1 1 1 1
%e A230025   1
%e A230025   1
%e A230025 The self-conjugate portion of p is
%e A230025   1 1 1 1
%e A230025   1 1
%e A230025   1
%e A230025   1
%e A230025 so that the number of outliers of p is 2.
%t A230025 ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; conjugatePartition[part_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[part]; selfConjugatePortion[list_] := ferrersMatrix[#]*ferrersMatrix[conjugatePartition[#]] &[list]; outliers[list_] := Count[Flatten[ferrersMatrix[#] - selfConjugatePortion[#] &[list]], 1]; a[n_] := Map[outliers, IntegerPartitions[n]]; t = Table[Count[a[n], k], {n, 1, 13}, {k, 0, n - 1}]
%t A230025 u = Flatten[t]
%t A230025 (* _Peter J. C. Moses_, Feb 21 2014 *)
%Y A230025 Cf. A000041, A366156.
%K A230025 nonn,tabl,easy
%O A230025 1,3
%A A230025 _Clark Kimberling_, Feb 23 2014