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