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A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.

%I A325232 #6 Apr 15 2019 01:38:03
%S A325232 2,3,6,10,18,27,44,64,97,138,200,276,390,528,724,968,1301,1712,2266,
%T A325232 2946,3842,4947,6372,8122,10362,13094,16544,20754,26010,32392,40308,
%U A325232 49876,61648,75845,93178,114006,139308,169586,206158,249814,302267,364664,439330
%N A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.
%H A325232 Giovanni Resta, <a href="/A325232/b325232.txt">Table of n, a(n) for n = 0..75</a>
%H A325232 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000533">St000533: The maximal number of non-attacking rooks on a Ferrers shape</a>
%F A325232 For n > 0, a(n) = Sum_{k > 0} A325227(n + k, k).
%e A325232 The a(0) = 1 through a(4) = 18 partitions:
%e A325232   ()   (2)   (3)    (4)     (5)
%e A325232   (1)  (11)  (22)   (32)    (33)
%e A325232        (21)  (31)   (41)    (42)
%e A325232              (111)  (221)   (51)
%e A325232              (211)  (321)   (222)
%e A325232              (311)  (411)   (322)
%e A325232                     (1111)  (331)
%e A325232                     (2111)  (421)
%e A325232                     (3111)  (511)
%e A325232                     (4111)  (2211)
%e A325232                             (3211)
%e A325232                             (4211)
%e A325232                             (5111)
%e A325232                             (11111)
%e A325232                             (21111)
%e A325232                             (31111)
%e A325232                             (41111)
%e A325232                             (51111)
%t A325232 nn=30;
%t A325232 mindif[ptn_]:=If[ptn=={},0,Total[ptn]-Min[Length[ptn],Max[ptn]]];
%t A325232 allip=Array[IntegerPartitions,2*nn+2,0,Join];
%t A325232 Table[Length[Select[allip,mindif[#]==n&]],{n,0,nn}]
%Y A325232 Number of times n appears in A325224.
%Y A325232 Cf. A051924, A257541, A263297, A325193, A325194, A325224, A325225, A325227.
%K A325232 nonn
%O A325232 0,1
%A A325232 _Gus Wiseman_, Apr 13 2019
%E A325232 More terms from _Giovanni Resta_, Apr 15 2019