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A279945 Irregular triangular array: t(n,k) = number of partitions of n having lexicographic difference set of size k; see Comments. 11
1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 6, 4, 1, 4, 10, 1, 6, 14, 1, 1, 8, 17, 4, 1, 8, 27, 6, 1, 6, 36, 13, 1, 13, 42, 21, 1, 7, 58, 35, 1, 10, 72, 52, 1, 15, 75, 84, 1, 1, 12, 106, 107, 5, 1, 9, 119, 159, 9, 1, 19, 142, 204, 19, 1, 10, 164, 283, 32, 1, 16, 199 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A partition P = [p(1), p(2), ..., p(k)] with p(1) >= p(2) >= ... >= p(k) has lexicographic difference set {0} union {|p(i) - p(i-1)|: 2 <= i <= k}. Column 2 is A049990, and the n-th row sum is A000041(n).

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..100

EXAMPLE

First 20 rows of array:

1

1    1

1    2

1    3    1

1    3    3

1    6    4

1    4    10

1    6    14    1

1    8    17    4

1    8    27    6

1    6    36    13

1    13   42    21

1    7    58    35

1    10   72    52

1    15   75    84    1

1    12   106   107   5

1    9    119   159   9

1    19   142   204   19

1    10   164   283   32

1    16   199   360   51

Row 5: the 7 partitions of 5 are shown here with difference sets:

partition  difference set     size

[5]          null              0

[4,1]        {3}               1

[3,2]        {1}               1

[3,1,1]      {0,2}             2

[2,2,1]      {0,1}             2

[2,1,1,1]    {0,1}             2

[1,1,1,1]    {0}               1

Row 5 of the array is 1 3 3, these being the number of 0's, 1's, 2's in the "size" column.

MATHEMATICA

p[n_] := IntegerPartitions[n]; z = 20;

t[n_, k_] := Length[DeleteDuplicates[Abs[Differences[p[n][[k]]]]]];

u[n_] := Table[t[n, k], {k, 1, PartitionsP[n]}];

v = Table[Count[u[n], h], {n, 1, z}, {h, 0, Max[u[n]]}]

TableForm[v] (* A279945 array *)

Flatten[v]   (* A279945 sequence *)

CROSSREFS

Cf. A000041, A049990.

Sequence in context: A244740 A088742 A256435 * A300322 A144220 A156826

Adjacent sequences:  A279942 A279943 A279944 * A279946 A279947 A279948

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 26 2016

STATUS

approved

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Last modified April 6 08:06 EDT 2020. Contains 333267 sequences. (Running on oeis4.)