A325357 - OEIS

%I #11 Mar 04 2021 03:18:49

%S 1,1,1,1,2,1,2,2,2,2,3,3,3,3,4,3,5,5,4,5,6,5,7,7,7,7,9,7,10,10,8,11,

%T 13,10,13,14,12,14,17,13,17,19,17,18,22,19,22,24,21,24,28,24,29,30,28,

%U 31,35,30,35,40,36

%N Number of integer partitions of n whose augmented differences are strictly increasing.

%C The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

%C The Heinz numbers of these partitions are given by A325395.

%H Fausto A. C. Cariboni, <a href="/A325357/b325357.txt">Table of n, a(n) for n = 0..2000</a>

%e The a(28) = 10 partitions:

%e   (28)

%e   (18,10)

%e   (17,11)

%e   (16,12)

%e   (15,13)

%e   (14,14)

%e   (12,10,6)

%e   (11,10,7)

%e   (10,10,8)

%e   (8,8,7,5)

%e For example, the augmented differences of (8,8,7,5) are (1,2,3,5), which are strictly increasing.

%t aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];

%t Table[Length[Select[IntegerPartitions[n],Less@@aug[#]&]],{n,0,30}]

%Y Cf. A000837, A007294, A049988, A098859, A325351, A325356, A325360, A325391, A325395.

%K nonn

%O 0,5

%A _Gus Wiseman_, Apr 23 2019