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Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.
8

%I #6 Apr 15 2019 01:38:03

%S 2,3,6,10,18,27,44,64,97,138,200,276,390,528,724,968,1301,1712,2266,

%T 2946,3842,4947,6372,8122,10362,13094,16544,20754,26010,32392,40308,

%U 49876,61648,75845,93178,114006,139308,169586,206158,249814,302267,364664,439330

%N 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 Giovanni Resta, <a href="/A325232/b325232.txt">Table of n, a(n) for n = 0..75</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000533">St000533: The maximal number of non-attacking rooks on a Ferrers shape</a>

%F For n > 0, a(n) = Sum_{k > 0} A325227(n + k, k).

%e The a(0) = 1 through a(4) = 18 partitions:

%e () (2) (3) (4) (5)

%e (1) (11) (22) (32) (33)

%e (21) (31) (41) (42)

%e (111) (221) (51)

%e (211) (321) (222)

%e (311) (411) (322)

%e (1111) (331)

%e (2111) (421)

%e (3111) (511)

%e (4111) (2211)

%e (3211)

%e (4211)

%e (5111)

%e (11111)

%e (21111)

%e (31111)

%e (41111)

%e (51111)

%t nn=30;

%t mindif[ptn_]:=If[ptn=={},0,Total[ptn]-Min[Length[ptn],Max[ptn]]];

%t allip=Array[IntegerPartitions,2*nn+2,0,Join];

%t Table[Length[Select[allip,mindif[#]==n&]],{n,0,nn}]

%Y Number of times n appears in A325224.

%Y Cf. A051924, A257541, A263297, A325193, A325194, A325224, A325225, A325227.

%K nonn

%O 0,1

%A _Gus Wiseman_, Apr 13 2019

%E More terms from _Giovanni Resta_, Apr 15 2019