cp's OEIS Frontend

A342531 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.

%I A342531 #7 Mar 25 2021 12:44:21
%S A342531 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 A342531 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 A342531 2,3,1,1,1,1,0,1,0,0
%N A342531 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 A342531 The maximal descent of an empty or singleton partition is considered to be 0.
%H A342531 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%e A342531 Triangle begins:
%e A342531 1
%e A342531 1 0
%e A342531 1 0 0
%e A342531 1 1 0 0
%e A342531 1 0 1 0 0
%e A342531 1 1 0 1 0 0
%e A342531 1 1 1 0 1 0 0
%e A342531 1 1 1 1 0 1 0 0
%e A342531 1 0 2 1 1 0 1 0 0
%e A342531 1 2 1 1 1 1 0 1 0 0
%e A342531 1 1 2 2 1 1 1 0 1 0 0
%e A342531 1 1 2 3 1 1 1 1 0 1 0 0
%e A342531 1 1 3 2 3 1 1 1 1 0 1 0 0
%e A342531 1 1 3 3 3 2 1 1 1 1 0 1 0 0
%e A342531 1 1 3 4 3 3 2 1 1 1 1 0 1 0 0
%e A342531 1 3 3 4 4 3 2 2 1 1 1 1 0 1 0 0
%e A342531 1 0 5 5 5 4 3 2 2 1 1 1 1 0 1 0 0
%e A342531 1 1 4 7 5 5 4 2 2 2 1 1 1 1 0 1 0 0
%e A342531 1 2 5 6 7 6 4 4 2 2 2 1 1 1 1 0 1 0 0
%e A342531 1 1 5 9 7 7 6 4 3 2 2 2 1 1 1 1 0 1 0 0
%e A342531 1 1 6 9 9 7 8 5 4 3 2 2 2 1 1 1 1 0 1 0 0
%e A342531 Row n = 15 counts the following strict partitions (empty columns indicated by dots, A..F = 10..15):
%e A342531   F  87     753   96    762   A5   A41   B4   B31  C3  C21  D2  .  E1  .  .
%e A342531      654    6432  852   843   861  9321  A32
%e A342531      54321  6531  7431  951   942
%e A342531                   7521  8421
%t A342531 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&If[Length[#]<=1,k==0,Max[Differences[Reverse[#]]]==k]&]],{n,0,15},{k,0,n}]
%Y A342531 The non-strict version is A238353.
%Y A342531 A000041 counts partitions (strict: A000009).
%Y A342531 A049980 counts strict partitions with equal differences.
%Y A342531 A325325 counts partitions with distinct differences (ranking: A325368).
%Y A342531 A325545 counts compositions with distinct differences.
%Y A342531 Cf. A003114, A003242, A005117, A034296, A049988, A238710, A342098, A342514.
%K A342531 nonn,tabl
%O A342531 0,39
%A A342531 _Gus Wiseman_, Mar 25 2021