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A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

%I A334440 #26 Sep 22 2023 08:45:50
%S A334440 0,1,1,1,1,2,1,1,1,2,2,1,1,2,2,2,2,2,1,1,1,2,2,1,3,2,2,2,2,1,1,2,2,2,
%T A334440 2,2,3,2,2,3,2,2,2,2,1,1,1,2,2,2,2,2,3,3,2,1,3,2,3,2,2,3,2,2,2,2,1,1,
%U A334440 2,2,2,2,1,3,2,2,3,3,2,2,3,3,3,3,2,2,3
%N A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.
%C A334440 The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.
%H A334440 Robert Price, <a href="/A334440/b334440.txt">Table of n, a(n) for n = 0..9295</a> (25 rows).
%H A334440 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A334440 a(n) = A001221(A334433(n)).
%e A334440 Triangle begins:
%e A334440   0
%e A334440   1
%e A334440   1 1
%e A334440   1 2 1
%e A334440   1 1 2 2 1
%e A334440   1 2 2 2 2 2 1
%e A334440   1 1 2 2 1 3 2 2 2 2 1
%e A334440   1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
%e A334440   1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1
%t A334440 Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]],{n,0,10}]
%Y A334440 Row lengths are A000041.
%Y A334440 The number of not necessarily distinct parts is A036043.
%Y A334440 The version for reversed partitions is A103921.
%Y A334440 Ignoring length (sum/lex) gives A103921 (also).
%Y A334440 a(n) is the number of distinct elements in row n of A334301.
%Y A334440 The maximum part of the same partition is A334441.
%Y A334440 Lexicographically ordered reversed partitions are A026791.
%Y A334440 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334440 Partitions in increasing-length colex order (sum/length/colex) are A036037.
%Y A334440 Graded reverse-lexicographically ordered partitions are A080577.
%Y A334440 Partitions counted by sum and number of distinct parts are A116608.
%Y A334440 Graded lexicographically ordered partitions are A193073.
%Y A334440 Partitions in colexicographic order (sum/colex) are A211992.
%Y A334440 Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
%Y A334440 Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.
%K A334440 nonn,tabf
%O A334440 0,6
%A A334440 _Gus Wiseman_, May 05 2020