cp's OEIS Frontend

A325356 Number of integer partitions of n whose augmented differences are weakly increasing.

%I A325356 #9 Mar 03 2021 19:28:59
%S A325356 1,1,2,2,3,3,4,3,6,5,5,6,8,6,10,9,8,10,13,10,15,14,13,15,21,15,19,21,
%T A325356 20,25,25,20,31,30,30,32,35,28,40,44,36,42,50,43,54,53,49,57,67,58,68,
%U A325356 66,66,78,84,71,86,92,82,99,109
%N A325356 Number of integer partitions of n whose augmented differences are weakly increasing.
%C A325356 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 A325356 The Heinz numbers of these partitions are given by A325394.
%H A325356 Fausto A. C. Cariboni, <a href="/A325356/b325356.txt">Table of n, a(n) for n = 0..1500</a>
%e A325356 The a(1) = 1 through a(8) = 6 partitions:
%e A325356   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A325356        (11)  (111)  (22)    (32)     (33)      (43)       (44)
%e A325356                     (1111)  (11111)  (222)     (1111111)  (53)
%e A325356                                      (111111)             (332)
%e A325356                                                           (2222)
%e A325356                                                           (11111111)
%e A325356 For example, the augmented differences of (6,6,5,3) are (1,2,3,3), which are weakly increasing, so (6,6,5,3) is counted under a(20).
%t A325356 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325356 Table[Length[Select[IntegerPartitions[n],OrderedQ[aug[#]]&]],{n,0,30}]
%Y A325356 Cf. A000837, A007294, A049988, A098859, A325350, A325351, A325354, A325357, A325358, A325360, A325394.
%K A325356 nonn
%O A325356 0,3
%A A325356 _Gus Wiseman_, Apr 23 2019