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A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal. 15
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 5, 2, 1, 0, 1, 5, 7, 5, 2, 1, 0, 1, 6, 11, 10, 5, 2, 1, 0, 1, 7, 15, 16, 10, 5, 2, 1, 0, 1, 8, 20, 24, 20, 10, 5, 2, 1, 0, 1, 9, 25, 36, 31, 20, 10, 5, 2, 1, 0, 1, 10, 32, 50, 50, 36, 20, 10, 5, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

A composition of n is a finite sequence of positive integers summing to n.

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

LINKS

Table of n, a(n) for n=0..77.

MathWorld, Unimodal Sequence

EXAMPLE

Triangle begins:

  1

  0  1

  0  1  1

  0  1  2  1

  0  1  3  2  1

  0  1  4  5  2  1

  0  1  5  7  5  2  1

  0  1  6 11 10  5  2  1

  0  1  7 15 16 10  5  2  1

  0  1  8 20 24 20 10  5  2  1

  0  1  9 25 36 31 20 10  5  2  1

  0  1 10 32 50 50 36 20 10  5  2  1

  0  1 11 38 67 73 59 36 20 10  5  2  1

Column n = 7 counts the following compositions:

  (7)  (16)  (115)  (1114)  (11113)  (111112)  (1111111)

       (25)  (124)  (1123)  (11122)  (211111)

       (34)  (133)  (1222)  (21112)

       (43)  (214)  (2113)  (22111)

       (52)  (223)  (2122)  (31111)

       (61)  (313)  (2212)

             (322)  (2221)

             (331)  (3112)

             (412)  (3211)

             (421)  (4111)

             (511)

MATHEMATICA

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], unimodQ[-#]&]], {n, 0, 10}, {k, 0, n}]

CROSSREFS

The case of partitions is A072233.

Dominated by A072704 (the non-negated version).

The strict case is A072705.

The case of constant compositions is A113704.

Row sums are A332578.

Unimodal compositions are A001523.

Unimodal normal sequences appear to be A007052.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Numbers whose negated unsorted prime signature is not unimodal are A332282.

Partitions whose negated run-lengths are unimodal are A332638.

Compositions whose negation is not unimodal are A332669.

Partitions whose negated 0-appended first differences are unimodal: A332728.

Cf. A011782, A107429, A227038, A332280, A332283, A332639, A332642, A332741, A332742, A332744, A332832, A332870.

Sequence in context: A256140 A321391 A244003 * A118344 A119270 A267109

Adjacent sequences:  A332667 A332668 A332669 * A332671 A332672 A332673

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Feb 29 2020

STATUS

approved

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Last modified January 18 15:35 EST 2021. Contains 340254 sequences. (Running on oeis4.)