login
Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.
32

%I #10 Apr 16 2019 18:32:10

%S 1,2,3,4,1,4,3,8,3,6,9,10,12,11,19,15,26,1,13,39,4,25,47,5,19,70,12,

%T 29,89,17,33,115,28,42,148,41,39,189,69,62,235,88,55,294,141,81,362,

%U 183,1,84,450,253,5,103,558,333,8,105,669,464,17,153,817,576,29

%N Irregular triangle read by rows with zeros removed where T(n,k) is the number of integer partitions of n with k distinct multiplicities, n > 0.

%C For example, the partition (32111) has multiplicities {1,1,3}, of which 2 are distinct, so is counted under T(8,2).

%H Alois P. Heinz, <a href="/A325242/b325242.txt">Rows n = 1..200, flattened</a>

%e Triangle begins:

%e 1

%e 2

%e 3

%e 4 1

%e 4 3

%e 8 3

%e 6 9

%e 10 12

%e 11 19

%e 15 26 1

%e 13 39 4

%e 25 47 5

%e 19 70 12

%e 29 89 17

%e 33 115 28

%e 42 148 41

%e 39 189 69

%e 62 235 88

%e 55 294 141

%e 81 362 183 1

%e Row n = 8 counts the following partitions:

%e (8) (332)

%e (44) (422)

%e (53) (611)

%e (62) (3221)

%e (71) (4211)

%e (431) (5111)

%e (521) (22211)

%e (2222) (32111)

%e (3311) (41111)

%e (11111111) (221111)

%e (311111)

%e (2111111)

%t DeleteCases[Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==k&]],{n,20},{k,n}],0,2]

%Y Row lengths are A056556. Row sums are A000041. Column k = 1 is A047966. Column k = 2 is A325243.

%Y Cf. A008284, A062770, A071625, A098859, A116608, A127002, A183558, A243978, A244515, A325244, A325268.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Apr 15 2019