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.

2, 3, 6, 10, 18, 27, 44, 64, 97, 138, 200, 276, 390, 528, 724, 968, 1301, 1712, 2266, 2946, 3842, 4947, 6372, 8122, 10362, 13094, 16544, 20754, 26010, 32392, 40308, 49876, 61648, 75845, 93178, 114006, 139308, 169586, 206158, 249814, 302267, 364664, 439330

Offset: 0

Author

Gus Wiseman, Apr 13 2019

Keywords

• nonn

Examples

			The a(0) = 1 through a(4) = 18 partitions:
  ()   (2)   (3)    (4)     (5)
  (1)  (11)  (22)   (32)    (33)
       (21)  (31)   (41)    (42)
             (111)  (221)   (51)
             (211)  (321)   (222)
             (311)  (411)   (322)
                    (1111)  (331)
                    (2111)  (421)
                    (3111)  (511)
                    (4111)  (2211)
                            (3211)
                            (4211)
                            (5111)
                            (11111)
                            (21111)
                            (31111)
                            (41111)
                            (51111)
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..75
• FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Crossrefs

Number of times n appears in A325224.

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

Programs

Mathematica

nn=30;
mindif[ptn_]:=If[ptn=={},0,Total[ptn]-Min[Length[ptn],Max[ptn]]];
allip=Array[IntegerPartitions,2*nn+2,0,Join];
Table[Length[Select[allip,mindif[#]==n&]],{n,0,nn}]

Formula

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

Extensions

More terms from Giovanni Resta, Apr 15 2019