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

Author

Gus Wiseman, Feb 19 2020

Keywords

• nonn

Comments

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

Examples

			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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Unimodal Sequence.

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.

Programs

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}]

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

Formula

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

Extensions

a(9) from Robert Price, Jun 19 2021

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