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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 run-lengths are unimodal.
+10
18
1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368
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.
EXAMPLE
The only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).
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[Length/@Split[#]]&]], {n, 0, 10}]
PROG
(PARI)
step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m, i]), s=vecsum(v)); M[p, ]+=vector(#M, i, s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M, m)); vector(n, i, vecsum(M[i, ]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[, 1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1, 1..n-m+2]; S+=concat(vector(m), desc(D, m))); S} \\ Andrew Howroyd, Dec 31 2020
CROSSREFS
Looking at the composition itself (not run-lengths) gives A001523. The case of partitions is A332280, with complement counted by A332281. The complement is counted by A332727. Unimodal normal sequences appear to be A007052. Non-unimodal compositions are A115981. 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 negated run-lengths are unimodal are A332578. Compositions whose negated run-lengths are not unimodal are A332669. Compositions whose run-lengths are weakly increasing are A332836. Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.
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 unimodal permutations of a multiset whose multiplicities are the prime indices of n.
+10
14
1, 1, 1, 2, 1, 3, 1, 4, 3, 4, 1, 6, 1, 5, 4, 8, 1, 9, 1, 8, 5, 6, 1, 12, 4, 7, 9, 10, 1, 12, 1, 16, 6, 8, 5, 18, 1, 9, 7, 16, 1, 15, 1, 12, 12, 10, 1, 24, 5, 16, 8, 14, 1, 27, 6, 20, 9, 11, 1, 24, 1, 12, 15, 32, 7, 18, 1, 16, 10, 20, 1, 36, 1, 13, 16, 18, 6
COMMENTS
This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(12) = 6 permutations:
{1,1,2,3}
{1,1,3,2}
{1,2,3,1}
{1,3,2,1}
{2,3,1,1}
{3,2,1,1}
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]], unimodQ]], {n, 0, 30}]
CROSSREFS
A less interesting version is A332288. The complement is counted by A332672. The opposite/negative version is A332741. Non-unimodal permutations are A059204. Partitions whose run-lengths are unimodal are A332280. Cf. A007052, A056239, A112798, A115981, A124010, A304660, A328509, A332283, A332578, A332638, A332671, A332742.
Number of compositions of n whose run-lengths are not unimodal.
+10
12
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
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.
EXAMPLE
The a(6) = 1 through a(8) = 8 compositions:
(11211) (11311) (11411)
(111211) (111311)
(112111) (112112)
(113111)
(211211)
(1111211)
(1112111)
(1121111)
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[Length/@Split[#]]&]], {n, 0, 10}]
CROSSREFS
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. 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.
Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.
+10
12
0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 0, 8, 0, 3, 7, 16, 0, 24, 0, 16, 12, 4, 0, 52, 16, 5, 81, 26, 0, 54, 0, 104, 18, 6, 31, 168, 0, 7, 25, 112, 0, 99, 0, 38, 201, 8, 0, 344, 65, 132, 33, 52, 0, 612, 52, 202, 42, 9, 0, 408, 0, 10, 411, 688, 80, 162, 0, 68, 52, 272
COMMENTS
This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
121 132 1212 1121 1132 11121 11212 1243
231 1221 1211 1213 11211 11221 1324
2121 1231 12111 12112 1342
1312 12121 1423
1321 12211 1432
2131 21121 2143
2311 21211 2314
3121 2341
2413
2431
3142
3241
3412
3421
4132
4231
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]], !unimodQ[#]&]], {n, 30}]
CROSSREFS
The complement of the non-negated version is counted by A332294. The non-negated version is A332672. The complement is counted by A332741. A less interesting version is A333146. Unimodal normal sequences are A007052. Non-unimodal normal sequences are A328509. Partitions with non-unimodal 0-appended first differences are A332284. Compositions whose negation is unimodal are A332578. Partitions with non-unimodal negated run-lengths are A332639. Numbers whose negated prime signature is not unimodal are A332642. Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.
Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.
+10
11
1, 1, 2, 3, 4, 5, 7, 8, 10, 13, 14, 17, 22, 24, 28, 34, 37, 43, 53, 56, 64, 76, 83, 93, 111, 117, 131, 153, 163, 182, 210, 225, 250, 284, 304, 332, 377, 401, 441, 497, 529, 576, 647, 687, 745, 830, 883, 955, 1062, 1127, 1216, 1339, 1422, 1532, 1684, 1779, 1914
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(8) = 10 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(321) (421) (332)
(111111) (2221) (431)
(1111111) (521)
(2222)
(11111111)
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 non-negated version is A332283. The non-negated complement is counted by A332284. The case of run-lengths (instead of differences) is A332638. The complement is counted by A332744. The Heinz numbers of partitions not in this class are A332287. Compositions whose negation is unimodal are A332578. Compositions whose run-lengths are unimodal are A332726.
Number of non-unimodal permutations of a multiset whose multiplicities are the prime indices of n.
+10
10
0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 6, 16, 0, 21, 0, 12, 10, 0, 0, 48, 16, 0, 81, 20, 0, 48, 0, 104, 15, 0, 30, 162, 0, 0, 21, 104, 0, 90, 0, 30, 198, 0, 0, 336, 65, 124, 28, 42, 0, 603, 50, 190, 36, 0, 0, 396, 0, 0, 405, 688, 77, 150, 0, 56, 45, 260, 0
COMMENTS
This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(n) permutations for n = 8, 9, 12, 15, 16:
213 1212 1213 11212 1324
312 2112 1312 12112 1423
2121 2113 12121 2134
2131 21112 2143
3112 21121 2314
3121 21211 2413
3124
3142
3214
3241
3412
4123
4132
4213
4231
4312
MATHEMATICA
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]], !unimodQ[#]&]], {n, 30}]
CROSSREFS
Positions of zeros are one and A001751. The complement is counted by A332294. A less interesting version is A332671. Non-unimodal permutations are A059204. Non-unimodal compositions are A115981. Non-unimodal normal sequences are A328509. Heinz numbers of partitions with non-unimodal run-lengths are A332282. Compositions whose negation is not unimodal are A332669. Cf. A007052, A008480, A056239, A112798, A124010, A181819, A181821, A332281, A332287, A332294, A332642, A332741.
Number of non-unimodal negated permutations of the multiset of prime indices of n.
+10
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 7, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 0, 8, 0, 0, 0
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
EXAMPLE
The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 96:
(121) (1121) (1212) (1132) (11212) (1232) (111121)
(1211) (1221) (1213) (11221) (1322) (111211)
(2121) (1231) (12112) (2132) (112111)
(1312) (12121) (2231) (121111)
(1321) (12211) (2312)
(2131) (21121) (2321)
(2311) (21211)
(3121)
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]]]];
Table[Length[Select[Permutations[primeMS[n]], !unimodQ[-#]&]], {n, 30}]
CROSSREFS
The non-negated version is A332671. A more interesting version is A332742. The complement is counted by A333145. Unimodal normal sequences are A007052. Compositions whose negation is unimodal are A332578. Partitions with unimodal negated run-lengths are A332638. Numbers with non-unimodal negated unsorted prime signature are A332642. Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332288, A332294, A332639, A332669, A332670, A332741.
Number of unimodal negated permutations of the multiset of prime indices of n.
+10
3
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).
Also the number divisors of n not divisible by the least prime factor of n. The other divisors are counted by A069157. - Gus Wiseman, Apr 12 2022
EXAMPLE
The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:
(1) (12) (122) (123) (1223) (12222) (1234) (12233)
(21) (212) (213) (2123) (21222) (2134) (21233)
(221) (312) (2213) (22122) (3124) (22133)
(321) (3122) (22212) (3214) (31223)
(3212) (22221) (4123) (32123)
(3221) (4213) (32213)
(4312) (33122)
(4321) (33212)
(33221)
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]]]];
Table[Length[Select[Permutations[primeMS[n]], unimodQ[-#]&]], {n, 30}]
CROSSREFS
The complementary divisors are counted by A069157. The non-negated version is A332288. A more interesting version is A332741. The complement is counted by A333146. A001523 counts unimodal compositions.
A007052 counts unimodal normal sequences.
A028233 gives the highest power of the least prime factor, quotient A028234.
A332578 counts compositions whose negation is unimodal.
A332638 counts partitions with unimodal negated run-lengths.
A332642 lists numbers with non-unimodal negated unsorted prime signature.
Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.
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