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A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.

%I A355522 #10 Jul 14 2022 09:34:50
%S A355522 2,2,1,3,1,1,2,3,1,1,4,3,2,1,1,2,6,3,2,1,1,4,6,6,2,2,1,1,3,10,6,5,2,2,
%T A355522 1,1,4,11,11,6,4,2,2,1,1,2,16,13,10,5,4,2,2,1,1,6,17,19,12,9,4,4,2,2,
%U A355522 1,1,2,24,24,18,11,8,4,4,2,2,1,1
%N A355522 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with maximal difference k, if singletons have maximal difference 0.
%C A355522 The triangle starts with n = 2, and k ranges from 0 to n - 2.
%H A355522 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A355522 Triangle begins:
%e A355522   2
%e A355522   2  1
%e A355522   3  1  1
%e A355522   2  3  1  1
%e A355522   4  3  2  1  1
%e A355522   2  6  3  2  1  1
%e A355522   4  6  6  2  2  1  1
%e A355522   3 10  6  5  2  2  1  1
%e A355522   4 11 11  6  4  2  2  1  1
%e A355522   2 16 13 10  5  4  2  2  1  1
%e A355522   6 17 19 12  9  4  4  2  2  1  1
%e A355522   2 24 24 18 11  8  4  4  2  2  1  1
%e A355522   4 27 34 22 17 10  7  4  4  2  2  1  1
%e A355522   4 35 39 33 20 15  9  7  4  4  2  2  1  1
%e A355522   5 39 56 39 30 19 14  8  7  4  4  2  2  1  1
%e A355522 For example, row n = 8 counts the following reversed partitions:
%e A355522   (8)         (233)      (35)      (125)    (26)    (116)  (17)
%e A355522   (44)        (1223)     (134)     (11114)  (1115)
%e A355522   (2222)      (11123)    (224)
%e A355522   (11111111)  (11222)    (1124)
%e A355522               (111122)   (1133)
%e A355522               (1111112)  (111113)
%t A355522 Table[Length[Select[Reverse/@IntegerPartitions[n], If[Length[#]==1,0,Max@@Differences[#]]==k&]],{n,2,15},{k,0,n-2}]
%Y A355522 Crossrefs found in the link are not repeated here.
%Y A355522 Leading terms are A000005.
%Y A355522 Row sums are A000041.
%Y A355522 Counts m such that A056239(m) = n and A286470(m) = k.
%Y A355522 This is a trimmed version of A238353, which extends to k = n.
%Y A355522 For minimum instead of maximum we have A238354.
%Y A355522 Ignoring singletons entirely gives A238710.
%Y A355522 A001522 counts partitions with a fixed point (unproved), ranked by A352827.
%Y A355522 A115720 and A115994 count partitions by their Durfee square.
%Y A355522 A279945 counts partitions by number of distinct differences.
%Y A355522 Cf. A064428, A091602, A179254, A238352, A239455, A286469, A325404, A355524, A355526, A355532.
%K A355522 nonn,tabl
%O A355522 2,1
%A A355522 _Gus Wiseman_, Jul 08 2022