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Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.
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%I #7 Mar 25 2021 12:44:21

%S 1,1,0,1,0,0,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,1,0,1,

%T 0,0,1,0,2,1,1,0,1,0,0,1,2,1,1,1,1,0,1,0,0,1,1,2,2,1,1,1,0,1,0,0,1,1,

%U 2,3,1,1,1,1,0,1,0,0

%N Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.

%C The maximal descent of an empty or singleton partition is considered to be 0.

%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 1

%e 1 0

%e 1 0 0

%e 1 1 0 0

%e 1 0 1 0 0

%e 1 1 0 1 0 0

%e 1 1 1 0 1 0 0

%e 1 1 1 1 0 1 0 0

%e 1 0 2 1 1 0 1 0 0

%e 1 2 1 1 1 1 0 1 0 0

%e 1 1 2 2 1 1 1 0 1 0 0

%e 1 1 2 3 1 1 1 1 0 1 0 0

%e 1 1 3 2 3 1 1 1 1 0 1 0 0

%e 1 1 3 3 3 2 1 1 1 1 0 1 0 0

%e 1 1 3 4 3 3 2 1 1 1 1 0 1 0 0

%e 1 3 3 4 4 3 2 2 1 1 1 1 0 1 0 0

%e 1 0 5 5 5 4 3 2 2 1 1 1 1 0 1 0 0

%e 1 1 4 7 5 5 4 2 2 2 1 1 1 1 0 1 0 0

%e 1 2 5 6 7 6 4 4 2 2 2 1 1 1 1 0 1 0 0

%e 1 1 5 9 7 7 6 4 3 2 2 2 1 1 1 1 0 1 0 0

%e 1 1 6 9 9 7 8 5 4 3 2 2 2 1 1 1 1 0 1 0 0

%e Row n = 15 counts the following strict partitions (empty columns indicated by dots, A..F = 10..15):

%e F 87 753 96 762 A5 A41 B4 B31 C3 C21 D2 . E1 . .

%e 654 6432 852 843 861 9321 A32

%e 54321 6531 7431 951 942

%e 7521 8421

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&If[Length[#]<=1,k==0,Max[Differences[Reverse[#]]]==k]&]],{n,0,15},{k,0,n}]

%Y The non-strict version is A238353.

%Y A000041 counts partitions (strict: A000009).

%Y A049980 counts strict partitions with equal differences.

%Y A325325 counts partitions with distinct differences (ranking: A325368).

%Y A325545 counts compositions with distinct differences.

%Y Cf. A003114, A003242, A005117, A034296, A049988, A238710, A342098, A342514.

%K nonn,tabl

%O 0,39

%A _Gus Wiseman_, Mar 25 2021