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A210950 Triangle read by rows: T(n,k) = number of parts in the k-th column of the partitions of n but with the partitions aligned to the right margin.

%I A210950 #27 Oct 22 2023 17:05:38
%S A210950 1,1,2,1,2,3,1,2,4,5,1,2,4,6,7,1,2,4,7,10,11,1,2,4,7,11,14,15,1,2,4,7,
%T A210950 12,17,21,22,1,2,4,7,12,18,25,29,30,1,2,4,7,12,19,28,36,41,42,1,2,4,7,
%U A210950 12,19,29,40,50,55,56,1,2,4,7,12,19,30,43
%N A210950 Triangle read by rows: T(n,k) = number of parts in the k-th column of the partitions of n but with the partitions aligned to the right margin.
%C A210950 Index of the first partition of n that has k parts, when the partitions of n are listed in reverse lexicographic order, as in Mathematica's IntegerPartitions[n]. - _Clark Kimberling_, Oct 16 2023
%F A210950 T(n,k) = Sum_{j=1..n} A210951(j,k).
%e A210950 For n = 6 the partitions of 6 aligned to the right margin look like this:
%e A210950 .
%e A210950 .                      6
%e A210950 .                  3 + 3
%e A210950 .                  4 + 2
%e A210950 .              2 + 2 + 2
%e A210950 .                  5 + 1
%e A210950 .              3 + 2 + 1
%e A210950 .              4 + 1 + 1
%e A210950 .          2 + 2 + 1 + 1
%e A210950 .          3 + 1 + 1 + 1
%e A210950 .      2 + 1 + 1 + 1 + 1
%e A210950 .  1 + 1 + 1 + 1 + 1 + 1
%e A210950 .
%e A210950 The number of parts in columns 1-6 are
%e A210950 .  1,  2,  4,  7, 10, 11, the same as the 6th row of triangle.
%e A210950 Triangle begins:
%e A210950   1;
%e A210950   1, 2;
%e A210950   1, 2, 3;
%e A210950   1, 2, 4, 5;
%e A210950   1, 2, 4, 6, 7;
%e A210950   1, 2, 4, 7, 10, 11;
%e A210950   1, 2, 4, 7, 11, 14, 15;
%e A210950   1, 2, 4, 7, 12, 17, 21, 22;
%e A210950   1, 2, 4, 7, 12, 18, 25, 29, 30;
%e A210950   1, 2, 4, 7, 12, 19, 28, 36, 41, 42;
%e A210950   1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56;
%e A210950   1, 2, 4, 7, 12, 19, 30, 43, 58, 70, 76, 77;
%t A210950 m[n_, k_] := Length[IntegerPartitions[n][[k]]]; c[n_] := PartitionsP[n];
%t A210950 t[n_, h_] := Select[Range[c[n]], m[n, #] == h &, 1];
%t A210950 Column[Table[t[n, h], {n, 1, 20}, {h, 1, n}]]
%t A210950  (* _Clark Kimberling_, Oct 16 2023 *)
%Y A210950 Mirror of A058399. Row sums give A006128. Right border gives A000041, n >= 1. Rows converge to A000070.
%Y A210950 Cf. A135010, A194714, A210945, A210951, A210952, A210953, A210970.
%K A210950 nonn,tabl
%O A210950 1,3
%A A210950 _Omar E. Pol_, Apr 22 2012