A328509 Number of non-unimodal sequences of length n covering an initial interval of positive integers.

0, 0, 0, 3, 41, 425, 4287, 45941, 541219, 7071501, 102193755, 1622448861, 28090940363, 526856206877, 10641335658891, 230283166014653, 5315654596751659, 130370766738143517, 3385534662263335179, 92801587315936355325, 2677687796232803000171, 81124824998464533181661

Offset

0,4

Comments

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..200

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Formula

a(n) = A000670(n) - A007052(n-1) for n > 0. - Andrew Howroyd, Jan 28 2024

Example

The a(3) = 3 sequences are (2,1,2), (2,1,3), (3,1,2).
The a(4) = 41 sequences:
  (1212)  (2113)  (2134)  (2413)  (3142)  (3412)
  (1213)  (2121)  (2143)  (3112)  (3212)  (4123)
  (1312)  (2122)  (2212)  (3121)  (3213)  (4132)
  (1323)  (2123)  (2213)  (3122)  (3214)  (4213)
  (1324)  (2131)  (2312)  (3123)  (3231)  (4231)
  (1423)  (2132)  (2313)  (3124)  (3241)  (4312)
  (2112)  (2133)  (2314)  (3132)  (3312)

Mathematica

allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Union@@Permutations/@allnorm[n], !unimodQ[#]&]], {n, 0, 5}]

Prog

(PARI) seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - (1 - 3*x + x^2)/(1 - 4*x + 2*x^2), -(n+1)) \\ Andrew Howroyd, Jan 28 2024

Crossrefs

Not requiring non-unimodality gives A000670.

The complement is counted by A007052.

The case where the negation is not unimodal either is A332873.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Unimodal compositions covering an initial interval are A227038.

Numbers whose unsorted prime signature is not unimodal are A332282.

Covering partitions with unimodal run-lengths are A332577.

Non-unimodal compositions covering an initial interval are A332743.

Cf. A060223, A255906, A332281, A332284, A332639, A332672, A332834, A332870.

Sequence in context: A322244 A181226 A159249 * A087544 A305667 A213378

Adjacent sequences: A328506 A328507 A328508 * A328510 A328511 A328512

Keyword

nonn

Author

Gus Wiseman, Feb 19 2020

Extensions

a(9) from Robert Price, Jun 19 2021

a(10) onwards from Andrew Howroyd, Jan 28 2024

Status

approved