A227038 -id:A227038 - cp's OEIS frontend

A332727 Number of compositions of n whose run-lengths are not unimodal.

0, 0, 0, 0, 0, 0, 1, 3, 8, 28, 74, 188, 468, 1120, 2596, 5944, 13324, 29437, 64288, 138929, 297442, 632074, 1333897, 2798352, 5840164, 12132638, 25102232, 51750419, 106346704, 217921161, 445424102, 908376235, 1848753273, 3755839591, 7617835520, 15428584567, 31207263000

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

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

			The a(6) = 1 through a(8) = 8 compositions:
  (11211)  (11311)   (11411)
           (111211)  (111311)
           (112111)  (112112)
                     (113111)
                     (211211)
                     (1111211)
                     (1112111)
                     (1121111)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not its run-lengths) gives A115981.
The case of partitions is A332281, with complement counted by A332280.
The complement is counted by A332726.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negation is not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A007052, A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332578, A332638, A332639, A332670, A332741, A332833.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[Length/@Split[#]]&]],{n,0,10}]

a(n) + A332726(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 14, 35, 83, 193, 417, 890, 1847, 3809, 7805, 15833, 32028, 64513, 129671, 260155, 521775, 1044982, 2092692, 4188168, 8381434, 16767650, 33544423, 67098683, 134213022, 268443023, 536912014, 1073846768, 2147720476, 4295440133, 8590833907

Offset: 0

Gus Wiseman, Mar 02 2020

nonn

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

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

			The a(5) = 1 through a(7) = 14 compositions:
  (212)  (213)   (1213)
         (312)   (1312)
         (1212)  (2113)
         (2112)  (2122)
         (2121)  (2131)
                 (2212)
                 (3112)
                 (3121)
                 (11212)
                 (12112)
                 (12121)
                 (21112)
                 (21121)
                 (21211)

MathWorld, Unimodal Sequence

Not requiring non-unimodality gives A107429.
Not requiring the covering condition gives A115981.
The complement is counted by A227038.
A version for partitions is A332579, with complement A332577.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
Cf. A007052, A072704, A072706, A332281, A332284, A332287, A332578, A332639, A332642, A332669, A332834, A332870.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&!unimodQ[#]&]],{n,0,10}]

For n > 0, a(n) = A107429(n) - A227038(n).

A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 28, 39, 55, 77, 107, 142, 194, 254, 332, 434, 563, 716, 919, 1162, 1464, 1841, 2305, 2857, 3555, 4383, 5394, 6617, 8099, 9859, 12006, 14551, 17600, 21236, 25574, 30688, 36809, 44007, 52527, 62574, 74430, 88304, 104675, 123799

Offset: 0

Gus Wiseman, Feb 27 2020

nonn

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

			The a(4) = 1 through a(9) = 17 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (2211)   (511)     (611)      (711)
                 (3111)   (3211)    (3221)     (3222)
                 (21111)  (4111)    (3311)     (4221)
                          (22111)   (4211)     (4311)
                          (31111)   (5111)     (5211)
                          (211111)  (22211)    (6111)
                                    (32111)    (32211)
                                    (41111)    (33111)
                                    (221111)   (42111)
                                    (311111)   (51111)
                                    (2111111)  (222111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Unimodal Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The complement is counted by A332728.
The non-negated version is A332284.
The strict case is A332579.
The case of run-lengths (instead of differences) is A332639.
The Heinz numbers of these partitions are A332832.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Numbers whose negated prime signature is not unimodal are A332642.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],!unimodQ[-Differences[Append[#,0]]]&]],{n,0,30}]

A072707 Number of non-unimodal compositions of n into distinct terms.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 24, 26, 46, 64, 100, 224, 276, 416, 590, 850, 1144, 2214, 2644, 3938, 5282, 7504, 9776, 13704, 21984, 27632, 38426, 51562, 69844, 91950, 123504, 159658, 246830, 303400, 416068, 540480, 730268, 933176, 1248110

Offset: 0

Henry Bottomley, Jul 04 2002

Also the number of compositions of n into distinct terms whose negation is not unimodal. - Gus Wiseman, Mar 05 2020

			a(6)=2 since 6 can be written as 2+1+3 or 3+1+2.
From _Gus Wiseman_, Mar 05 2020: (Start)
The a(6) = 2 through a(9) = 6 strict compositions:
  (2,1,3)  (2,1,4)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (3,1,4)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (4,2,3)
                             (5,1,3)
                             (6,1,2)
(End)

Eric Weisstein's World of Mathematics, Unimodal Sequence

The complement is counted by A072706.
The non-strict version is A115981.
The case where the negation is not unimodal either is A332874.
Unimodal compositions are A001523.
Strict compositions are A032020.
Non-unimodal permutations are A059204.
A triangle for strict unimodal compositions is A072705.
Non-unimodal sequences covering an initial interval are A328509.
Numbers whose prime signature is not unimodal are A332282.
Strict partitions whose 0-appended differences are not unimodal are A332286.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Non-unimodal compositions covering an initial interval are A332743.
Cf. A007052, A072704, A107429, A227038, A329398, A332281, A332284, A332285, A332579, A332870.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!unimodQ[#]&]],{n,0,16}] (* Gus Wiseman, Mar 05 2020 *)

a(n) = A032020(n) - A072706(n) = Sum_{k} A059204(k) * A060016(n, k).

A072705 Triangle of number of unimodal compositions of n into exactly k distinct terms.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 4, 0, 0, 0, 1, 6, 4, 0, 0, 0, 0, 1, 6, 8, 0, 0, 0, 0, 0, 1, 8, 12, 0, 0, 0, 0, 0, 0, 1, 8, 16, 8, 0, 0, 0, 0, 0, 0, 1, 10, 20, 8, 0, 0, 0, 0, 0, 0, 0, 1, 10, 28, 16, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 32, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 40, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jul 04 2002

Also the number of compositions of n into exactly k distinct terms whose negation is unimodal. - Gus Wiseman, Mar 06 2020

			Rows start: 1; 1,0; 1,2,0; 1,2,0,0; 1,4,0,0,0; 1,4,4,0,0,0; 1,6,4,0,0,0,0; 1,6,8,0,0,0,0,0; etc. T(6,3)=4 since 6 can be written as 1+2+3, 1+3+2, 2+3+1, or 3+2+1 but not 2+1+3 or 3+1+2.
From _Gus Wiseman_, Mar 06 2020: (Start)
Triangle begins:
  1
  1  0
  1  2  0
  1  2  0  0
  1  4  0  0  0
  1  4  4  0  0  0
  1  6  4  0  0  0  0
  1  6  8  0  0  0  0  0
  1  8 12  0  0  0  0  0  0
  1  8 16  8  0  0  0  0  0  0
  1 10 20  8  0  0  0  0  0  0  0
  1 10 28 16  0  0  0  0  0  0  0  0
  1 12 32 24  0  0  0  0  0  0  0  0  0
  1 12 40 40  0  0  0  0  0  0  0  0  0  0
  1 14 48 48 16  0  0  0  0  0  0  0  0  0  0
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A060016, A072574, A072704. Row sums are A072706.
Column k = 2 is A052928.
Unimodal compositions are A001523.
Unimodal sequences covering an initial interval are A007052.
Strict compositions are A032020.
Non-unimodal strict compositions are A072707.
Unimodal compositions covering an initial interval are A227038.
Numbers whose prime signature is not unimodal are A332282.
Cf. A059204, A107429, A115981, A329398, A332285, A332286, A332578, A332870, A332874.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i)*ceil(2^(i-1)), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i > n, 0, x*b[n-i, i-1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i]* Ceiling[2^(i-1)], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
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}],UnsameQ@@#&&unimodQ[#]&]],{n,12},{k,n}] (* Gus Wiseman, Mar 06 2020 *)

T(n,k) = 2^(k-1)*A060016(n,k) = T(n-k,k)+2*T(n-k,k-1) [starting with T(0,0)=0, T(0,1)=0 and T(n,1)=1 for n>0].

A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

12, 20, 24, 28, 36, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 165, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 195, 196, 198

Offset: 1

Gus Wiseman, Mar 02 2020

nonn

First differs from A065201 in having 165.
First differs from A316597 in having 36.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   90: {1,2,2,3}
For example, 60 is the Heinz number of (3,2,1,1), with negated 0-appended first-differences (1,1,0,1), which are not unimodal, so 60 is in the sequence.

MathWorld, Unimodal Sequence
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

The non-negated version is A332287.
The version for of run-lengths (instead of differences) is A332642.
The enumeration of these partitions by sum is A332744.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Heinz numbers of partitions with non-unimodal run-lengths are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Cf. A059204, A227038, A332284, A332285, A332286, A332578, A332638, A332639, A332670, A332725, A332728.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Select[Range[100],!unimodQ[Differences[Prepend[primeMS[#],0]]]&]

A332874 Number of strict compositions of n that are neither unimodal nor is their negation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 20, 30, 50, 150, 180, 290, 420, 630, 860, 1828, 2168, 3326, 4514, 6530, 8576, 12188, 20096, 25314, 35576, 48062, 65592, 86752, 117222, 152060, 237590, 292346, 402798, 524596, 711270, 910606, 1221204, 1554382, 2044460, 2927124

Offset: 0

Gus Wiseman, Mar 04 2020

nonn

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

A composition of n is a finite sequence of positive integers summing to n. It is strict if there are not repeated parts.

			The a(10) = 10 through a(12) = 20 compositions:
  (1,3,2,4)  (1,3,2,5)  (1,3,2,6)
  (1,4,2,3)  (1,5,2,3)  (1,4,2,5)
  (2,1,4,3)  (2,1,5,3)  (1,5,2,4)
  (2,3,1,4)  (2,3,1,5)  (1,6,2,3)
  (2,4,1,3)  (2,5,1,3)  (2,1,5,4)
  (3,1,4,2)  (3,1,5,2)  (2,1,6,3)
  (3,2,4,1)  (3,2,5,1)  (2,3,1,6)
  (3,4,1,2)  (3,5,1,2)  (2,4,1,5)
  (4,1,3,2)  (5,1,3,2)  (2,5,1,4)
  (4,2,3,1)  (5,2,3,1)  (2,6,1,3)
                        (3,1,6,2)
                        (3,2,6,1)
                        (3,6,1,2)
                        (4,1,5,2)
                        (4,2,5,1)
                        (4,5,1,2)
                        (5,1,4,2)
                        (5,2,4,1)
                        (6,1,3,2)
                        (6,2,3,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence

The non-strict version for unsorted prime signature is A332643.
The non-strict version is A332870.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Compositions with neither weakly increasing nor weakly decreasing run-lengths are A332833.
Compositions with weakly increasing or weakly decreasing run-lengths are A332835.
Cf. A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332640, A332641, A332745, A332746, A332831, A332834.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!unimodQ[#]&&!unimodQ[-#]&]],{n,0,20}]

seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=4, n, (k! - 2^k + 2)*polcoef(p,k,y)), -(n+1))} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=4} (k! - 2^k + 2) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

Terms a(21) and beyond from Andrew Howroyd, Apr 16 2021

A333149 Number of strict compositions of n that are neither increasing nor decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 4, 8, 12, 38, 42, 72, 98, 150, 298, 372, 542, 760, 1070, 1428, 2600, 3120, 4550, 6050, 8478, 10976, 15220, 23872, 29950, 41276, 55062, 74096, 97148, 129786, 167256, 256070, 314454, 429338, 556364, 749266, 955746, 1275016, 1618054

Offset: 0

Gus Wiseman, May 16 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.

			The a(6) = 4 through a(9) = 12 compositions:
  (1,3,2)  (1,4,2)  (1,4,3)  (1,5,3)
  (2,1,3)  (2,1,4)  (1,5,2)  (1,6,2)
  (2,3,1)  (2,4,1)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (2,5,1)  (2,4,3)
                    (3,1,4)  (2,6,1)
                    (3,4,1)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (3,4,2)
                             (3,5,1)
                             (4,2,3)
                             (5,1,3)
                             (6,1,2)

Eric Weisstein's World of Mathematics, Unimodal Sequence

The non-strict case is A332834.
The complement is counted by A333147.
Strict partitions are A000009.
Strict compositions are A032020.
Non-unimodal strict compositions are A072707.
Strict partitions with increasing or decreasing run-lengths are A333190.
Strict compositions with increasing or decreasing run-lengths are A333191.
Unimodal compositions are A001523, with strict case A072706.
Cf. A059204, A115981, A227038, A329398, A332745, A332746, A332831, A332833, A332835, A332874, A333150, A333192.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!Greater@@#&&!Less@@#&]],{n,0,10}]

a(n) = A032020(n) - 2*A000009(n) + 1.

A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 14, 14, 5, 0, 1, 30, 45, 32, 8, 0, 1, 62, 124, 131, 65, 13, 0, 1, 126, 315, 438, 323, 128, 21, 0, 1, 254, 762, 1305, 1270, 747, 243, 34, 0, 1, 510, 1785, 3612, 4346, 3370, 1629, 452, 55

Offset: 0

Gus Wiseman, Mar 02 2020

In other words, parts of subsequent, non-successive blocks are increasing.

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    3
    0    1   14   14    5
    0    1   30   45   32    8
    0    1   62  124  131   65   13
    0    1  126  315  438  323  128   21
    0    1  254  762 1305 1270  747  243   34
    ...
Row n = 4 counts the following ordered set partitions:
  {1234}  {1}{234}  {1}{2}{34}  {1}{2}{3}{4}
          {12}{34}  {1}{23}{4}  {1}{2}{4}{3}
          {123}{4}  {12}{3}{4}  {1}{3}{2}{4}
          {124}{3}  {1}{24}{3}  {2}{1}{3}{4}
          {13}{24}  {12}{4}{3}  {2}{1}{4}{3}
          {134}{2}  {1}{3}{24}
          {14}{23}  {13}{2}{4}
          {2}{134}  {1}{34}{2}
          {23}{14}  {1}{4}{23}
          {234}{1}  {2}{1}{34}
          {24}{13}  {2}{13}{4}
          {3}{124}  {2}{14}{3}
          {34}{12}  {23}{1}{4}
          {4}{123}  {3}{12}{4}

An apparently related triangle is A056242.
Column k = n - 1 is A332724.
Row sums are A332872, which appears to be A007052 shifted right once.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Cf. A072704, A097805, A107429, A227038, A332280, A332283, A332288, A332577.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==k&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,5},{k,0,n}]

A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

Original entry on oeis.org

0, 0, 1, 6, 14, 32, 65, 128, 243, 452, 826, 1490, 2659, 4704, 8261, 14418, 25030, 43252, 74437, 127648, 218199, 371920, 632306, 1072486, 1815239, 3066432, 5170825, 8705118, 14632958, 24562952, 41177801, 68947520, 115313979, 192656924, 321554986, 536191418

Offset: 0

Gus Wiseman, Mar 03 2020

nonn

In other words, parts of not-immediately-subsequent blocks are increasing.

			The a(2) = 1 through a(4) = 14 ordered set partitions:
  {{1,2}}  {{1},{2,3}}  {{1},{2},{3,4}}
           {{1,2},{3}}  {{1},{2,3},{4}}
           {{1,3},{2}}  {{1,2},{3},{4}}
           {{2},{1,3}}  {{1},{2,4},{3}}
           {{2,3},{1}}  {{1,2},{4},{3}}
           {{3},{1,2}}  {{1},{3},{2,4}}
                        {{1,3},{2},{4}}
                        {{1},{3,4},{2}}
                        {{1},{4},{2,3}}
                        {{2},{1},{3,4}}
                        {{2},{1,3},{4}}
                        {{2},{1,4},{3}}
                        {{2,3},{1},{4}}
                        {{3},{1,2},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Column k = n - 1 of A332673, which has row-sums A332872.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal normal sequences are A328509.
Cf. A000045, A001286, A001629, A056242, A227038, A332577.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==n-1&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,8}]

\\ here b(n) is A001629(n).
b(n) = {((n+1)*fibonacci(n-1) + (n-1)*fibonacci(n+1))/5}
a(n) = {if(n==0, 0, b(n) + 4*b(n-1) + b(n-2))} \\ Andrew Howroyd, Apr 17 2021

From Andrew Howroyd, Apr 17 2021: (Start)
a(n) = A001629(n) + 4*A001629(n+1) + A001629(n+2) for n > 0.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 4.
G.f.: x*(1 + 4*x + x^2)/(1 - x - x^2)^2.
(End)

Terms a(9) and beyond from Andrew Howroyd, Apr 17 2021

cp's OEIS Frontend

A332727 Number of compositions of n whose run-lengths are not unimodal.

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A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.

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A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

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A072707 Number of non-unimodal compositions of n into distinct terms.

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A072705 Triangle of number of unimodal compositions of n into exactly k distinct terms.

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A332832 Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.

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A332874 Number of strict compositions of n that are neither unimodal nor is their negation.

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