A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.

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

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

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Formula

G.f.: A(x,y) = 1 + Sum_{j>0} y*x^j/((1 - y*x^j)*Product_{k>j} (1 - y*x^k)^2). - Andrew Howroyd, Jan 11 2024

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}]

Prog

(PARI) T(n)={[Vecrev(p) | p<-Vec(1 + sum(j=1, n, y*x^j/((1-y*x^j) * prod(k=j+1, n-j, 1 - y*x^k + O(x*x^(n-j)))^2)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

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: A244003 A369738 A360763 * A118344 A343138 A119270

Adjacent sequences: A332667 A332668 A332669 * A332671 A332672 A332673

Keyword

nonn,tabl

Author

Gus Wiseman, Feb 29 2020

Status

approved