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T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows.
1

%I #11 Dec 24 2022 22:25:01

%S 1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,5,1,2,1,1,1,7,1,3,1,1,1,1,10,1,5,

%T 1,2,1,1,1,13,1,7,1,3,2,1,1,1,17,1,10,1,5,3,2,1,1,1,21,1,13,1,7,6,3,1,

%U 1,1,1,26,1,17,1,10,10,5,1,3,1,1,1,31,1

%N T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows.

%C Column n consists solely of 1's if and only if n = 1 or n is a prime. Column 4: A033638.

%H Clark Kimberling, <a href="/A237513/b237513.txt">Table of n, a(n) for n = 1..500</a>

%e Triangle begins:

%e 1

%e 1 ... 1

%e 1 ... 1 ... 1

%e 1 ... 1 ... 1 ... 3

%e 1 ... 1 ... 1 ... 3 ... 1

%e 1 ... 1 ... 1 ... 5 ... 1 ... 2

%e 1 ... 1 ... 1 ... 7 ... 1 ... 3 ... 1

%e 1 ... 1 ... 1 .. 10 ... 1 ... 5 ... 1 ... 2

%e 1 ... 1 ... 1 .. 13 ... 1 ... 7 ... 1 ... 3 ... 2

%e 1 ... 1 ... 1 .. 17 ... 1 .. 10 ... 1 ... 5 ... 3 ... 2

%e 1 ... 1 ... 1 .. 21 ... 1 .. 13 ... 1 ... 7 ... 6 ... 3 ... 1

%e 1 ... 1 ... 1 .. 26 ... 1 .. 17 ... 1 .. 10 .. 10 ... 5 ... 1 ... 3

%e The dimensions of maximal horizontal rectangles for the partitions of 6 are 1 X 6, 1 X 5, 2 X 2, 1 X 4, 2 X 3, 2 X 2, 1 X 3, 2 X 2, 2 X 2, 1 X 2, 1 X 1; the numbers of nodes are the products 6, 5, 4, 4, 6, 4, 3, 4, 4, 2, 1; counting the occurrences of 1,2,3,4,5,6 gives 1,1,1,5,1,2, which is row 6 of the array.

%t durfeeSquare[part_] := Max[Map[Min, Transpose[{#, Range[Length[#]]}] &[part]]]; hRectangle[part_] := {#, part[[#]]} &[durfeeSquare[part]]; Map[Last[Transpose[Tally[Sort[Map[Times @@ hRectangle[#] &, IntegerPartitions[#]]]]]] &, Range[20]] (* _Peter J. C. Moses_, Feb 10 2014 *)

%Y Cf. A033638, A000041, A033638.

%K nonn,tabl,easy

%O 1,10

%A _Clark Kimberling_, Feb 08 2014