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Number of compositions of n whose negation is unimodal.
+10
43
1, 1, 2, 4, 7, 13, 21, 36, 57, 91, 140, 217, 323, 485, 711, 1039, 1494, 2144, 3032, 4279, 5970, 8299, 11438, 15708, 21403, 29065, 39218, 52725, 70497, 93941, 124562, 164639, 216664, 284240, 371456, 484004, 628419, 813669, 1050144, 1351757, 1734873, 2221018, 2835613
COMMENTS
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.
FORMULA
G.f.: 1 + Sum_{j>0} x^j/((1 - x^j)*(Product_{k>j} 1 - x^k)^2). - Andrew Howroyd, Mar 01 2020 a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Mar 01 2020
EXAMPLE
The a(1) = 1 through a(5) = 13 compositions:
(1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(211) (113)
(1111) (122)
(212)
(221)
(311)
(1112)
(2111)
(11111)
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], unimodQ[-#]&]], {n, 0, 10}]
nmax = 50; CoefficientList[Series[1 + Sum[x^j*(1 - x^j)/Product[1 - x^k, {k, j, nmax - j}]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 01 2020 *)
PROG
(PARI) seq(n)={Vec(1 + sum(j=1, n, x^j/((1-x^j)*prod(k=j+1, n-j, 1 - x^k + O(x*x^(n-j)))^2)))} \\ Andrew Howroyd, Mar 01 2020
CROSSREFS
Dominated by A001523 (unimodal compositions). The case that is unimodal also is A329398. The complement is counted by A332669. Unimodal normal sequences appear to be A007052. Non-unimodal compositions are A115981. Non-unimodal normal sequences are A328509. Partitions whose run-lengths are unimodal are A332280. Partitions whose negated run-lengths are unimodal are A332638. Numbers whose unsorted prime signature is not unimodal are A332642. Partitions whose negated 0-appended differences are unimodal are A332728. Cf. A011782, A072704, A107429, A227038, A332282, A332283, A332639, A332741, A332742, A332744, A332832, A332870.
Number of compositions of n whose negation is not unimodal.
+10
37
0, 0, 0, 0, 1, 3, 11, 28, 71, 165, 372, 807, 1725, 3611, 7481, 15345, 31274, 63392, 128040, 257865, 518318, 1040277, 2085714, 4178596, 8367205, 16748151, 33515214, 67056139, 134147231, 268341515, 536746350, 1073577185, 2147266984, 4294683056, 8589563136, 17179385180
COMMENTS
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.
FORMULA
a(n) + A332578(n) = 2^(n - 1) for n > 0.
EXAMPLE
The a(4) = 1 through a(6) = 11 compositions:
(121) (131) (132)
(1121) (141)
(1211) (231)
(1131)
(1212)
(1221)
(1311)
(2121)
(11121)
(11211)
(12111)
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], !unimodQ[-#]&]], {n, 0, 10}]
CROSSREFS
The complement is counted by A332578. The version for run-lengths of partitions is A332639. The version for unsorted prime signature is A332642. The version for 0-appended first-differences of partitions is A332744. The case that is not unimodal either is A332870. Non-unimodal permutations are A059204. Non-unimodal compositions are A115981. Non-unimodal normal sequences are A328509. Numbers whose unsorted prime signature is not unimodal are A332282. A triangle for compositions with unimodal negation is A332670.
Number of integer partitions of n whose negated run-lengths are unimodal.
+10
28
1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 52, 70, 91, 118, 151, 195, 246, 310, 388, 484, 600, 743, 909, 1113, 1359, 1650, 1996, 2409, 2895, 3471, 4156, 4947, 5885, 6985, 8260, 9751, 11503, 13511, 15857, 18559, 21705, 25304, 29499, 34259, 39785, 46101, 53360, 61594
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(8) = 21 partitions:
(8) (44) (2222)
(53) (332) (22211)
(62) (422) (32111)
(71) (431) (221111)
(521) (3311) (311111)
(611) (4211) (2111111)
(5111) (41111) (11111111)
Missing from this list is only (3221).
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], unimodQ[-Length/@Split[#]]&]], {n, 0, 30}]
CROSSREFS
The non-negated version is A332280. The complement is counted by A332639. The Heinz numbers of partitions not in this class are A332642. The case of 0-appended differences (instead of run-lengths) is A332728. Partitions whose run lengths are not unimodal are A332281. Heinz numbers of partitions with non-unimodal run-lengths are A332282. Compositions whose negation is unimodal are A332578. Compositions whose run-lengths are unimodal are A332726.
Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms.
+10
23
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 7, 5, 1, 1, 6, 12, 12, 9, 6, 1, 1, 7, 16, 20, 16, 11, 7, 1, 1, 8, 21, 30, 28, 20, 13, 8, 1, 1, 9, 27, 42, 45, 36, 24, 15, 9, 1, 1, 10, 33, 58, 68, 60, 44, 28, 17, 10, 1, 1, 11, 40, 77, 98, 95, 75, 52, 32, 19, 11, 1
FORMULA
G.f. with initial column 1, 0, 0, ...: 1 + Sum_{n>=1} (t*x^n / ( ( Product_{k=1..n-1} (1 - t*x^k)^2 ) * (1 - t*x^n) ) ). - Joerg Arndt, Oct 01 2017
EXAMPLE
Rows start:
01: [1]
02: [1, 1]
03: [1, 2, 1]
04: [1, 3, 3, 1]
05: [1, 4, 5, 4, 1]
06: [1, 5, 8, 7, 5, 1]
07: [1, 6, 12, 12, 9, 6, 1]
08: [1, 7, 16, 20, 16, 11, 7, 1]
09: [1, 8, 21, 30, 28, 20, 13, 8, 1]
10: [1, 9, 27, 42, 45, 36, 24, 15, 9, 1]
...
T(6,3)=8 since 6 can be written as 1+1+4, 1+2+3, 1+3+2, 1+4+1, 2+2+2, 2+3+1, 3+2+1, or 4+1+1 but not 2+1+3 or 3+1+2.
MAPLE
b:= proc(n, i) option remember; local q; `if`(i>n, 0,
`if`(irem(n, i, 'q')=0, x^q, 0) +expand(
add(b(n-i*j, i+1)*(j+1)*x^j, j=0..n/i)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
MATHEMATICA
b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i ] == 0, x^Quotient[n, i], 0] + Expand[ Sum[b[n-i*j, i+1]*(j+1)*x^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]]; Table[T[n], {n, 1, 12}] // 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}], unimodQ]], {n, 0, 10}, {k, 0, n}] (* Gus Wiseman, Mar 06 2020 *)
PROG
(PARI) \\ starting for n=0, with initial column 1, 0, 0, ...:
N=25; x='x+O('x^N);
T=Vec(1 + sum(n=1, N, t*x^n / ( prod(k=1, n-1, (1 - t*x^k)^2 ) * (1 - t*x^n) ) ) )
for(r=1, #T, print(Vecrev(T[r])) ); \\ Joerg Arndt, Oct 01 2017
CROSSREFS
Dominates A332670 (the version for negated compositions). The case of constant compositions is A113704. Unimodal sequences covering an initial interval are A007052. Partitions whose run-lengths are unimodal are A332280.
Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.
+10
19
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 25, 30, 36, 40, 45, 54, 59, 68, 79, 86, 96, 112, 121, 135, 155, 168, 188, 214, 230, 253, 284, 308, 337, 380, 407, 445, 497, 533, 580, 645, 689, 748, 828, 885, 956, 1053, 1124, 1212, 1330, 1415, 1519, 1665, 1771
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(1) = 1 through a(9) = 8 partitions:
1 11 21 211 221 321 2221 3221 3321
111 1111 2111 2211 3211 22211 22221
11111 21111 22111 32111 32211
111111 211111 221111 222111
1111111 2111111 321111
11111111 2211111
21111111
111111111
MATHEMATICA
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[IntegerPartitions[n], normQ[#]&&unimodQ[Length/@Split[#]]&]], {n, 0, 30}]
CROSSREFS
Not requiring unimodality gives A000009. A version for compositions is A227038. Not requiring the partition to cover an initial interval gives A332280. The complement is counted by A332579. Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.
Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.
+10
16
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
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.
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
Dominated by A072704 (the non-negated version). The case of constant compositions is A113704. 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.
Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.
+10
14
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 7, 8, 10, 14, 19, 22, 30, 36, 43, 56, 69, 80, 101, 121, 141, 172, 202, 234, 282, 332, 384, 452, 527, 602, 706, 815, 929, 1077, 1236, 1403, 1615, 1842, 2082, 2379, 2702, 3044, 3458, 3908, 4388, 4963, 5589, 6252
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also the number of strict integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.
EXAMPLE
The a(10) = 1 through a(16) = 7 partitions:
33211 332111 3321111 333211 433211 443211 443221
33211111 3332111 4332111 3333211
332111111 33321111 4432111
3321111111 33322111
43321111
333211111
33211111111
MATHEMATICA
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[IntegerPartitions[n], normQ[#]&&!unimodQ[Length/@Split[#]]&]], {n, 0, 30}]
CROSSREFS
The complement is counted by A332577. Not requiring the partition to cover an initial interval gives A332281. A version for compositions is A332743. Partitions covering an initial interval of positive integers are A000009. Non-unimodal permutations are A059204. Non-unimodal compositions are A115981. Non-unimodal normal sequences are A328509. Numbers whose prime signature is not unimodal are A332282. Partitions whose 0-appended first differences are unimodal are A332283. Compositions whose negated run-lengths are not unimodal are A332727. Cf. A007052, A100883, A107429, A227038, A332280, A332284, A332638, A332639, A332640, A332671, A332672, A332728.
Neither the unsorted prime signature of a(n) nor the negated unsorted prime signature of a(n) is unimodal.
+10
11
2100, 3300, 3900, 4200, 4410, 5100, 5700, 6468, 6600, 6900, 7644, 7800, 8400, 8700, 9300, 9996, 10200, 10500, 10780, 10890, 11100, 11172, 11400, 12300, 12740, 12900, 12936, 13200, 13230, 13524, 13800, 14100, 15210, 15246, 15288, 15600, 15900, 16500, 16660
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
EXAMPLE
The sequence of terms together with their prime indices begins:
2100: {1,1,2,3,3,4}
3300: {1,1,2,3,3,5}
3900: {1,1,2,3,3,6}
4200: {1,1,1,2,3,3,4}
4410: {1,2,2,3,4,4}
5100: {1,1,2,3,3,7}
5700: {1,1,2,3,3,8}
6468: {1,1,2,4,4,5}
6600: {1,1,1,2,3,3,5}
6900: {1,1,2,3,3,9}
7644: {1,1,2,4,4,6}
7800: {1,1,1,2,3,3,6}
8400: {1,1,1,1,2,3,3,4}
8700: {1,1,2,3,3,10}
9300: {1,1,2,3,3,11}
9996: {1,1,2,4,4,7}
10200: {1,1,1,2,3,3,7}
10500: {1,1,2,3,3,3,4}
10780: {1,1,3,4,4,5}
10890: {1,2,2,3,5,5}
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Select[Range[10000], !unimodQ[Last/@FactorInteger[#]]&&!unimodQ[-Last/@FactorInteger[#]]&]
CROSSREFS
Not requiring non-unimodal negation gives A332282. These are the Heinz numbers of the partitions counted by A332640. Not requiring non-unimodality gives A332642. The case of compositions is A332870. Non-unimodal permutations are A059204. Non-unimodal compositions are A115981. Unsorted prime signature is A124010. Non-unimodal normal sequences are A328509. Partitions whose 0-appended first differences are unimodal are A332283, with Heinz numbers the complement of A332287. Compositions whose negation is unimodal are A332578. Compositions whose negation is not unimodal are A332669. Partitions whose 0-appended first differences are not unimodal are A332744, with Heinz numbers A332832. Numbers whose signature is neither increasing nor decreasing are A332831. Cf. A007052, A056239, A072704, A112798, A242031, A242414, A332280, A332281, A332288, A332294, A332639, A332728, A332742.
Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.
+10
9
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
COMMENTS
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
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).
MATHEMATICA
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}]
CROSSREFS
The complement is counted by A332728. The non-negated version is A332284. The case of run-lengths (instead of differences) is A332639. The Heinz numbers of these partitions are A332832. 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.
Heinz numbers of integer partitions whose negated first differences (assuming the last part is zero) are not unimodal.
+10
7
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
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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]]]&]
CROSSREFS
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. 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.
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