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A193073 Triangle in which n-th row lists all partitions of n, in graded lexicographical ordering. 40
1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 3, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 1, 1, 1, 3, 2, 1, 3, 3, 4, 1, 1, 4, 2, 5, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The partitions of the integer n are sorted in lexicographical order (cf. link: sums are written with terms in decreasing order, then they are sorted in lexicographical (increasing) order), i.e., as [1,1,...,1], [2,1,...,1], [2,2,...], ..., [n].

LINKS

Alois P. Heinz, Rows n = 1..19, flattened

OEIS Wiki, Orderings of partitions.

Wikiversity, Lexicographic and colexicographic order

EXAMPLE

First five rows are:

[[1]]

[[1, 1], [2]]

[[1, 1, 1], [2, 1], [3]]

[[1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]]

[[1, 1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1], [3, 1, 1], [3, 2], [4, 1], [5]]

From Gus Wiseman, May 08 2020: (Start)

The sequence of all partitions begins:

  ()           (2,2,1)        (5,1)            (5,2)

  (1)          (3,1,1)        (6)              (6,1)

  (1,1)        (3,2)          (1,1,1,1,1,1,1)  (7)

  (2)          (4,1)          (2,1,1,1,1,1)    (1,1,1,1,1,1,1,1)

  (1,1,1)      (5)            (2,2,1,1,1)      (2,1,1,1,1,1,1)

  (2,1)        (1,1,1,1,1,1)  (2,2,2,1)        (2,2,1,1,1,1)

  (3)          (2,1,1,1,1)    (3,1,1,1,1)      (2,2,2,1,1)

  (1,1,1,1)    (2,2,1,1)      (3,2,1,1)        (2,2,2,2)

  (2,1,1)      (2,2,2)        (3,2,2)          (3,1,1,1,1,1)

  (2,2)        (3,1,1,1)      (3,3,1)          (3,2,1,1,1)

  (3,1)        (3,2,1)        (4,1,1,1)        (3,2,2,1)

  (4)          (3,3)          (4,2,1)          (3,3,1,1)

  (1,1,1,1,1)  (4,1,1)        (4,3)            (3,3,2)

  (2,1,1,1)    (4,2)          (5,1,1)          (4,1,1,1,1)

The triangle with partitions shown as Heinz numbers (A334434) begins:

    1

    2

    4   3

    8   6   5

   16  12   9  10   7

   32  24  18  20  15  14  11

   64  48  36  27  40  30  25  28  21  22  13

  128  96  72  54  80  60  45  50  56  42  35  44  33  26  17

(End)

MATHEMATICA

row[n_] := Flatten[Reverse[Reverse /@ SplitBy[IntegerPartitions[n], Length] ], 1]; Array[row, 19] // Flatten (* Jean-Fran├žois Alcover, Dec 05 2016 *)

lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];

Join@@Table[Sort[IntegerPartitions[n], lexsort], {n, 0, 8}] (* Gus Wiseman, May 08 2020 *)

PROG

(PARI) A193073_row(n)=concat(vecsort(apply(P->Vec(vecsort(P, , 4)), partitions(n)))) \\ The two vecsort() are needed since the PARI function (version >= 2.7.1) yields the partitions in Abramowitz-Stegun order: sorted by increasing length, decreasing largest part, then lex order, with parts in increasing order. - M. F. Hasler, Jun 04 2018 [replaced older code from Jul 12 2015]

(Sage)

def p(n, i):

    if n==0 or i==1: return [[1]*n]

    T = [[i] + x for x in p(n-i, i)] if i<=n else []

    return p(n, i-1) + T

A193073 = lambda n: p(n, n)

for n in (1..5): print(A193073(n)) # Peter Luschny, Aug 07 2015

CROSSREFS

See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).

See A036037 for graded colexicographic ordering.

See A080576 for the Maple (graded reflected lexicographic) ordering.

See A080577 for the Mathematica (graded reverse lexicographic) ordering.

See A228100 for the Fenner-Loizou (binary tree) ordering.

A006128 gives row lengths.

Row n has A000041(n) partitions.

The version for reversed (weakly increasing) partitions is A026791.

Lengths of these partitions appear to be A049085.

Taking colex instead of lex gives A211992.

The generalization to compositions is A228351.

Sorting partitions by Heinz number gives A296150.

The length-sensitive refinement is A334301.

The Heinz numbers of these partitions are A334434.

Cf. A066099, A129129, A185974, A228531, A334302, A334433, A334437, A334439.

Sequence in context: A002951 A331287 A093993 * A228100 A211992 A182937

Adjacent sequences:  A193070 A193071 A193072 * A193074 A193075 A193076

KEYWORD

nonn,tabf

AUTHOR

M. F. Hasler, Jul 15 2011

STATUS

approved

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Last modified September 29 12:15 EDT 2020. Contains 337431 sequences. (Running on oeis4.)