A227038
Number of (weakly) unimodal compositions of n where all parts 1, 2, ..., m appear where m is the largest part.
Original entry on oeis.org
1, 1, 1, 3, 4, 7, 13, 19, 30, 44, 71, 98, 147, 205, 294, 412, 575, 783, 1077, 1456, 1957, 2634, 3492, 4627, 6082, 7980, 10374, 13498, 17430, 22451, 28767, 36806, 46803, 59467, 75172, 94839, 119285, 149599, 187031, 233355, 290340, 360327, 446222, 551251, 679524, 835964, 1026210
Offset: 0
There are a(8) = 30 such compositions of 8:
01: [ 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 2 ]
03: [ 1 1 1 1 1 2 1 ]
04: [ 1 1 1 1 2 1 1 ]
05: [ 1 1 1 1 2 2 ]
06: [ 1 1 1 2 1 1 1 ]
07: [ 1 1 1 2 2 1 ]
08: [ 1 1 1 2 3 ]
09: [ 1 1 1 3 2 ]
10: [ 1 1 2 1 1 1 1 ]
11: [ 1 1 2 2 1 1 ]
12: [ 1 1 2 2 2 ]
13: [ 1 1 2 3 1 ]
14: [ 1 1 3 2 1 ]
15: [ 1 2 1 1 1 1 1 ]
16: [ 1 2 2 1 1 1 ]
17: [ 1 2 2 2 1 ]
18: [ 1 2 2 3 ]
19: [ 1 2 3 1 1 ]
20: [ 1 2 3 2 ]
21: [ 1 3 2 1 1 ]
22: [ 1 3 2 2 ]
23: [ 2 1 1 1 1 1 1 ]
24: [ 2 2 1 1 1 1 ]
25: [ 2 2 2 1 1 ]
26: [ 2 2 3 1 ]
27: [ 2 3 1 1 1 ]
28: [ 2 3 2 1 ]
29: [ 3 2 1 1 1 ]
30: [ 3 2 2 1 ]
From _Gus Wiseman_, Mar 05 2020: (Start)
The a(1) = 1 through a(6) = 13 compositions:
(1) (11) (12) (112) (122) (123)
(21) (121) (221) (132)
(111) (211) (1112) (231)
(1111) (1121) (321)
(1211) (1122)
(2111) (1221)
(11111) (2211)
(11112)
(11121)
(11211)
(12111)
(21111)
(111111)
(End)
Cf.
A001523 (unimodal compositions),
A001522 (smooth unimodal compositions with first and last part 1),
A001524 (unimodal compositions such that each up-step is by at most 1 and first part is 1).
Organizing by length rather than sum gives
A007052.
The complement is counted by
A332743.
The case of run-lengths of partitions is
A332577, with complement
A332579.
Compositions covering an initial interval are
A107429.
Non-unimodal compositions are
A115981.
-
b:= proc(n,i) option remember;
`if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
add(b(n-i*j, i+1)*(j+1), j=1..n/i))
end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 26 2014
-
b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 1, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)
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}] (* Gus Wiseman, Mar 05 2020 *)
A332280
Number of integer partitions of n with unimodal run-lengths.
Original entry on oeis.org
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 97, 129, 166, 215, 273, 352, 439, 557, 692, 865, 1066, 1325, 1614, 1986, 2413, 2940, 3546, 4302, 5152, 6207, 7409, 8862, 10523, 12545, 14814, 17562, 20690, 24397, 28615, 33645, 39297, 46009, 53609, 62504, 72581, 84412
Offset: 0
The a(10) = 41 partitions (A = 10) are:
(A) (61111) (4321) (3211111)
(91) (55) (43111) (31111111)
(82) (541) (4222) (22222)
(811) (532) (42211) (222211)
(73) (5311) (421111) (2221111)
(721) (5221) (4111111) (22111111)
(7111) (52111) (3331) (211111111)
(64) (511111) (3322) (1111111111)
(631) (442) (331111)
(622) (4411) (32221)
(6211) (433) (322111)
Missing from this list is only (33211).
The complement is counted by
A332281.
Heinz numbers of these partitions are the complement of
A332282.
Taking 0-appended first-differences instead of run-lengths gives
A332283.
Unimodal normal sequences are
A007052.
Numbers whose unsorted prime signature is unimodal are
A332288.
Cf.
A007052,
A025065,
A072706,
A100883,
A115981,
A227038,
A317086,
A328509,
A329398,
A332284,
A332285,
A332294,
A332578,
A332579.
-
b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
end:
a:= n-> b(n$2, 0, true):
seq(a(n), n=0..65); # Alois P. Heinz, Feb 20 2020
-
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}]
(* Second program: *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, j, t && j >= m], {j, 1, Min[If[t, Infinity, m], n/i]}] + b[n, i - 1, m, t]]];
a[n_] := b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
A332281
Number of integer partitions of n whose run-lengths are not unimodal.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 10, 16, 24, 33, 51, 70, 100, 137, 189, 250, 344, 450, 597, 778, 1019, 1302, 1690, 2142, 2734, 3448, 4360, 5432, 6823, 8453, 10495, 12941, 15968, 19529, 23964, 29166, 35525, 43054, 52173, 62861, 75842, 91013, 109208
Offset: 0
The a(10) = 1 through a(15) = 10 partitions:
(33211) (332111) (44211) (44311) (55211) (44322)
(3321111) (333211) (433211) (55311)
(442111) (443111) (443211)
(33211111) (3332111) (533211)
(4421111) (552111)
(332111111) (4332111)
(4431111)
(33321111)
(44211111)
(3321111111)
The complement is counted by
A332280.
The Heinz numbers of these partitions are
A332282.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Non-unimodal normal sequences are
A328509.
Cf.
A007052,
A025065,
A072706,
A100883,
A332283,
A332284,
A332286,
A332287,
A332579,
A332638,
A332640,
A332641,
A332642.
-
b:= proc(n, i, m, t) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1, j, t and j>=m),
j=1..min(`if`(t, [][], m), n/i))+b(n, i-1, m, t)))
end:
a:= n-> combinat[numbpart](n)-b(n$2, 0, true):
seq(a(n), n=0..65); # Alois P. Heinz, Feb 20 2020
-
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}]
(* Second program: *)
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, j, t && j >= m], {j, 1, Min[If[t, Infinity, m], n/i]}] + b[n, i - 1, m, t]]];
a[n_] := PartitionsP[n] - b[n, n, 0, True];
a /@ Range[0, 65] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
A332639
Number of integer partitions of n whose negated run-lengths are not unimodal.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 10, 17, 25, 36, 51, 75, 102, 143, 192, 259, 346, 462, 599, 786, 1014, 1309, 1670, 2133, 2686, 3402, 4258, 5325, 6623, 8226, 10134, 12504, 15328, 18779, 22878, 27870, 33762, 40916, 49349, 59457, 71394, 85679, 102394
Offset: 0
The a(8) = 1 through a(13) = 10 partitions:
(3221) (4221) (5221) (4331) (4332) (5332)
(32221) (6221) (5331) (6331)
(42221) (7221) (8221)
(322211) (43221) (43321)
(52221) (53221)
(322221) (62221)
(422211) (332221)
(422221)
(522211)
(3222211)
The version for normal sequences is
A328509.
The non-negated complement is
A332280.
The non-negated version is
A332281.
The complement is counted by
A332638.
The case that is not unimodal either is
A332640.
The Heinz numbers of these partitions are
A332642.
The generalization to run-lengths of compositions is
A332727.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Compositions whose negation is not unimodal are
A332669.
Cf.
A007052,
A025065,
A100883,
A181819,
A332282,
A332578,
A332579,
A332641,
A332670,
A332671,
A332726,
A332742,
A332744.
-
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}]
A332284
Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 18, 28, 42, 62, 86, 123, 168, 226, 306, 411, 534, 704, 908, 1165, 1492, 1898, 2384, 3011, 3758, 4673, 5799, 7168, 8792, 10804, 13192, 16053, 19505, 23633, 28497, 34367, 41283, 49470, 59188, 70675, 84113, 100048, 118689, 140533
Offset: 0
The a(6) = 1 through a(11) = 18 partitions:
(2211) (331) (431) (441) (541) (551)
(22111) (3311) (4311) (3322) (641)
(22211) (32211) (3331) (4331)
(221111) (33111) (4411) (4421)
(222111) (33211) (5411)
(2211111) (42211) (33221)
(43111) (33311)
(222211) (44111)
(322111) (52211)
(331111) (322211)
(2221111) (332111)
(22111111) (422111)
(431111)
(2222111)
(3221111)
(3311111)
(22211111)
(221111111)
The complement is counted by
A332283.
The Heinz numbers of these partitions are
A332287.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Non-unimodal normal sequences appear to be
A328509.
Partitions with non-unimodal run-lengths are
A332281.
Heinz numbers of partitions with non-unimodal run-lengths are
A332282.
-
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,30}]
A332577
Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.
Original entry on oeis.org
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
Offset: 0
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
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.
-
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}]
A332287
Heinz numbers of integer partitions whose first differences (assuming the last part is zero) are not unimodal.
Original entry on oeis.org
36, 50, 70, 72, 98, 100, 108, 140, 144, 154, 180, 182, 196, 200, 216, 225, 242, 250, 252, 280, 286, 288, 294, 300, 308, 324, 338, 350, 360, 363, 364, 374, 392, 396, 400, 418, 429, 432, 441, 442, 450, 462, 468, 484, 490, 494, 500, 504, 507, 540, 550, 560, 561
Offset: 1
The sequence of terms together with their prime indices begins:
36: {1,1,2,2}
50: {1,3,3}
70: {1,3,4}
72: {1,1,1,2,2}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
154: {1,4,5}
180: {1,1,2,2,3}
182: {1,4,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
216: {1,1,1,2,2,2}
225: {2,2,3,3}
242: {1,5,5}
250: {1,3,3,3}
252: {1,1,2,2,4}
280: {1,1,1,3,4}
For example, the prime indices of 70 with 0 appended are (4,3,1,0), with differences (-1,-2,-1), which is not unimodal, so 70 belongs to the sequence.
The enumeration of these partitions by sum is
A332284.
Not assuming the last part is zero gives
A332725.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Non-unimodal normal sequences are
A328509.
Partitions with non-unimodal run-lengths are
A332281.
Cf.
A001523,
A007052,
A332280,
A332282,
A332283,
A332285,
A332286,
A332288,
A332294,
A332579,
A332639,
A332642.
-
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[1000],!unimodQ[Differences[Append[Reverse[primeMS[#]],0]]]&]
A332286
Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857
Offset: 0
The a(8) = 1 through a(18) = 7 partitions:
(431) . (541) (641) (651) (652) (752) (762) (862)
(5421) (751) (761) (861) (871)
(5431) (851) (6531) (961)
(6431) (7431) (6532)
(6521) (7521) (6541)
(7621)
(8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).
Partitions covering an initial interval are (also)
A000009.
The complement is counted by
A332285.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Non-unimodal normal sequences are
A328509.
Partitions with non-unimodal run-lengths are
A332281.
Normal partitions whose run-lengths are not unimodal are
A332579.
Cf.
A007052,
A011782,
A025065,
A072706,
A227038,
A332282,
A332283,
A332286,
A332287,
A332288,
A332577,
A332638,
A332642,
A332743.
-
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!unimodQ[Differences[Append[#,0]]]]&]],{n,0,30}]
A332640
Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 17, 29, 44, 66, 92, 138, 187, 266, 359, 492, 649, 877, 1140, 1503, 1938, 2517, 3202, 4111, 5175, 6563, 8209, 10297, 12763, 15898, 19568, 24152, 29575, 36249, 44090, 53737, 65022, 78752, 94873, 114294
Offset: 0
The a(14) = 1 through a(18) = 12 partitions:
(433211) (533211) (443221) (544211) (544311)
(4332111) (633211) (733211) (553221)
(5332111) (4333211) (644211)
(43321111) (6332111) (833211)
(53321111) (4432221)
(433211111) (5333211)
(5442111)
(7332111)
(43332111)
(63321111)
(533211111)
(4332111111)
For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).
Looking only at the original run-lengths gives
A332281.
Looking only at the negated run-lengths gives
A332639.
The Heinz numbers of these partitions are
A332643.
The complement is counted by
A332746.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Partitions with unimodal run-lengths are
A332280.
Partitions whose negated run-lengths are unimodal are
A332638.
Run-lengths and negated run-lengths are not both unimodal:
A332641.
Compositions whose negation is not unimodal are
A332669.
Run-lengths and negated run-lengths are both unimodal:
A332745.
Cf.
A007052,
A025065,
A100883,
A181819,
A328509,
A332282,
A332284,
A332577,
A332578,
A332579,
A332642,
A332726,
A332727.
-
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[#]]&&!unimodQ[-Length/@Split[#]]&]],{n,0,30}]
A332641
Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 14, 22, 33, 48, 69, 96, 136, 184, 248, 330, 443, 574, 756, 970, 1252, 1595, 2040, 2558, 3236, 4041, 5054, 6256, 7781, 9547, 11782, 14394, 17614, 21423, 26083, 31501, 38158, 45930, 55299, 66262, 79477, 94803, 113214
Offset: 0
The a(8) = 1 through a(13) = 14 partitions:
(3221) (4221) (5221) (4331) (4332) (5332)
(32221) (6221) (5331) (6331)
(33211) (42221) (7221) (8221)
(322211) (43221) (43321)
(332111) (44211) (44311)
(52221) (53221)
(322221) (62221)
(422211) (332221)
(3321111) (333211)
(422221)
(442111)
(522211)
(3222211)
(33211111)
The complement is counted by
A332745.
The Heinz numbers of these partitions are
A332831.
The case of run-lengths of compositions is
A332833.
Partitions whose run-lengths are weakly increasing are
A100883.
Partitions whose run-lengths are weakly decreasing are
A100882.
Partitions whose run-lengths are not unimodal are
A332281.
Partitions whose negated run-lengths are not unimodal are
A332639.
Non-unimodal permutations are
A059204.
Non-unimodal compositions are
A115981.
Partitions with unimodal run-lengths are
A332280.
Partitions whose negated run-lengths are unimodal are
A332638.
Compositions whose negation is not unimodal are
A332669.
The case of run-lengths of compositions is
A332833.
Compositions that are neither increasing nor decreasing are
A332834.
Cf.
A025065,
A181819,
A328509,
A332282,
A332284,
A332577,
A332578,
A332579,
A332640,
A332642,
A332726,
A332727,
A332742,
A332835.
-
Table[Length[Select[IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]
Showing 1-10 of 14 results.
Comments