The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A133121

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A133121

Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.
(history; published version)

#26 by Susanna Cuyler at Wed Jan 23 19:59:24 EST 2019

STATUS

proposed

approved

#25 by Gus Wiseman at Wed Jan 23 18:54:34 EST 2019

STATUS

editing

proposed

#24 by Gus Wiseman at Wed Jan 23 18:53:35 EST 2019

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Length[#]-Length[Union[#]]==k&]], {n, 0, 15}, {k, 0, n}] (* _augmented version, _Gus Wiseman_, Jan 23 2019 *)

#23 by Gus Wiseman at Wed Jan 23 05:46:35 EST 2019

EXAMPLE

1

1 0

1 1 0

2 0 1 0

2 2 0 1 0

3 2 1 0 1 0

4 2 3 1 0 1 0

5 4 2 2 1 0 1 0

6 6 3 3 2 1 0 1 0

8 7 5 4 2 2 1 0 1 0

10 8 10 3 5 2 2 1 0 1 0

12 13 8 9 4 4 2 2 1 0 1 0

15 15 14 10 8 5 4 2 2 1 0 1 0

18 21 15 16 8 9 4 4 2 2 1 0 1 0

22 25 23 17 17 7 10 4 4 2 2 1 0 1 0

27 30 32 21 19 16 8 9 4 4 2 2 1 0 1 0

CROSSREFS

Cf. A000009, A090858.

Cf. A000009, A090858, A100471, A116608.

#22 by Gus Wiseman at Wed Jan 23 05:45:08 EST 2019

EXAMPLE

From Gus Wiseman, Jan 23 2019: (Start)

It is possible to augment the triangle to cover the n = 0 and k = n cases, giving:

1

1 0

1 1 0

2 0 1 0

2 2 0 1 0

3 2 1 0 1 0

4 2 3 1 0 1 0

5 4 2 2 1 0 1 0

6 6 3 3 2 1 0 1 0

8 7 5 4 2 2 1 0 1 0

10 8 10 3 5 2 2 1 0 1 0

12 13 8 9 4 4 2 2 1 0 1 0

15 15 14 10 8 5 4 2 2 1 0 1 0

18 21 15 16 8 9 4 4 2 2 1 0 1 0

22 25 23 17 17 7 10 4 4 2 2 1 0 1 0

27 30 32 21 19 16 8 9 4 4 2 2 1 0 1 0

Row seven {5, 4, 2, 2, 1, 0, 1, 0} counts the following integer partitions (empty columns not shown).

(7) (322) (2221) (22111) (211111) (1111111)

(43) (331) (4111) (31111)

(52) (511)

(61) (3211)

(421)

(End)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Length[#]-Length[Union[#]]==k&]], {n, 0, 15}, {k, 0, n}] (* Gus Wiseman, Jan 23 2019 *)

CROSSREFS

Row sums are A000041. Row polynomials evaluated at -1 are A268498. Row polynomials evaluated at 2 are A006951.

Cf. A100471, A116608.

STATUS

approved

editing

#21 by Bruno Berselli at Sat Jan 23 08:33:22 EST 2016

STATUS

proposed

approved

#20 by Jean-François Alcover at Sat Jan 23 07:40:54 EST 2016

STATUS

editing

proposed

#19 by Jean-François Alcover at Sat Jan 23 07:40:49 EST 2016

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^If[j == 0, 0, j-1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[ Coefficient[p, x, i], {i, 0, n - 1}]][b[n, n]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)

STATUS

approved

editing

#18 by Bruno Berselli at Sun Aug 23 10:27:39 EDT 2015

STATUS

reviewed

approved

#17 by Joerg Arndt at Sun Aug 23 08:56:30 EDT 2015

STATUS

proposed

reviewed