cp's OEIS Frontend

A325357 Number of integer partitions of n whose augmented differences are strictly increasing.

%I A325357 #11 Mar 04 2021 03:18:49
%S A325357 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 A325357 13,10,13,14,12,14,17,13,17,19,17,18,22,19,22,24,21,24,28,24,29,30,28,
%U A325357 31,35,30,35,40,36
%N A325357 Number of integer partitions of n whose augmented differences are strictly increasing.
%C A325357 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 A325357 The Heinz numbers of these partitions are given by A325395.
%H A325357 Fausto A. C. Cariboni, <a href="/A325357/b325357.txt">Table of n, a(n) for n = 0..2000</a>
%e A325357 The a(28) = 10 partitions:
%e A325357   (28)
%e A325357   (18,10)
%e A325357   (17,11)
%e A325357   (16,12)
%e A325357   (15,13)
%e A325357   (14,14)
%e A325357   (12,10,6)
%e A325357   (11,10,7)
%e A325357   (10,10,8)
%e A325357   (8,8,7,5)
%e A325357 For example, the augmented differences of (8,8,7,5) are (1,2,3,5), which are strictly increasing.
%t A325357 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325357 Table[Length[Select[IntegerPartitions[n],Less@@aug[#]&]],{n,0,30}]
%Y A325357 Cf. A000837, A007294, A049988, A098859, A325351, A325356, A325360, A325391, A325395.
%K A325357 nonn
%O A325357 0,5
%A A325357 _Gus Wiseman_, Apr 23 2019