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

0,1

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

Formula

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

Example

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)

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}]

Crossrefs

Number of times n appears in A325224.

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

Sequence in context: A193200 A026779 A081028 * A208479 A065441 A075531

Adjacent sequences: A325229 A325230 A325231 * A325233 A325234 A325235

Keyword

nonn

Author

Gus Wiseman, Apr 13 2019

Extensions

More terms from Giovanni Resta, Apr 15 2019

Status

approved