A333190
Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 26, 29, 39, 49, 50, 68, 80, 92, 109, 129, 142, 181, 201, 227, 262, 317, 343, 404, 456, 516, 589, 677, 742, 870, 949, 1077, 1207, 1385, 1510, 1704, 1895, 2123, 2352, 2649, 2877, 3261, 3571, 3966, 4363, 4873, 5300, 5914, 6466
Offset: 0
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (2111) (411) (511) (422)
(11111) (3111) (2221) (611)
(21111) (4111) (2222)
(111111) (22111) (5111)
(31111) (22211)
(211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
The generalization to compositions is
A333191.
Partitions with distinct run-lengths are
A098859.
Partitions with strictly increasing run-lengths are
A100471.
Partitions with strictly decreasing run-lengths are
A100881.
Partitions with weakly decreasing run-lengths are
A100882.
Partitions with weakly increasing run-lengths are
A100883.
Partitions with unimodal run-lengths are
A332280.
Partitions whose run-lengths are not increasing nor decreasing are
A332641.
Compositions whose run-lengths are unimodal or co-unimodal are
A332746.
Compositions that are neither increasing nor decreasing are
A332834.
Strictly increasing or strictly decreasing compositions are
A333147.
Compositions with strictly increasing run-lengths are
A333192.
Numbers with strictly increasing prime multiplicities are
A334965.
-
Table[Length[Select[IntegerPartitions[n],Or[Less@@Length/@Split[#],Greater@@Length/@Split[#]]&]],{n,0,30}]
A337482
Number of compositions of n that are neither strictly increasing nor weakly decreasing.
Original entry on oeis.org
0, 0, 0, 0, 2, 7, 18, 45, 101, 219, 461, 957, 1957, 3978, 8036, 16182, 32506, 65202, 130642, 261601, 523598, 1047709, 2096062, 4192946, 8386912, 16775117, 33551832, 67105663, 134213789, 268430636, 536865013, 1073734643, 2147474910, 4294956706, 8589921771
Offset: 0
The a(4) = 2 through a(4) = 18 compositions:
(112) (113) (114)
(121) (122) (132)
(131) (141)
(212) (213)
(1112) (231)
(1121) (312)
(1211) (1113)
(1122)
(1131)
(1212)
(1221)
(1311)
(2112)
(2121)
(11112)
(11121)
(11211)
(12111)
A128422 counts only the case of length 3.
A001523 counts unimodal compositions, with complement counted by
A115981.
A332745/
A332835 count partitions/compositions with weakly increasing or weakly decreasing run-lengths.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!GreaterEqual@@#&]],{n,0,15}]
A321773
Number of compositions of n into parts with distinct multiplicities and with exactly three parts.
Original entry on oeis.org
1, 3, 6, 4, 9, 9, 10, 12, 15, 13, 18, 18, 19, 21, 24, 22, 27, 27, 28, 30, 33, 31, 36, 36, 37, 39, 42, 40, 45, 45, 46, 48, 51, 49, 54, 54, 55, 57, 60, 58, 63, 63, 64, 66, 69, 67, 72, 72, 73, 75, 78, 76, 81, 81, 82, 84, 87, 85, 90, 90, 91, 93, 96, 94, 99, 99
Offset: 3
From _Gus Wiseman_, Nov 11 2020: (Start)
Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
111 112 113 114 115 116 117 118 119
121 122 141 133 161 144 181 155
211 131 222 151 224 171 226 191
212 411 223 233 225 244 227
221 232 242 252 262 272
311 313 323 333 334 335
322 332 414 343 344
331 422 441 424 353
511 611 522 433 434
711 442 443
622 515
811 533
551
722
911
(End)
A235451 counts 3-part compositions with distinct run-lengths
A001399(n-6) counts 3-part compositions in the complement.
A261982 counts non-strict compositions of any length.
A032020 counts strict compositions.
A242771 counts triples that are not strictly increasing.
-
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* Gus Wiseman, Nov 11 2020 *)
A332873
Number of non-unimodal, non-co-unimodal sequences of length n covering an initial interval of positive integers.
Original entry on oeis.org
0, 0, 0, 0, 22, 340, 3954, 44716, 536858, 7056252, 102140970, 1622267196, 28090317226, 526854073564, 10641328363722, 230283141084220, 5315654511587498, 130370766447282204, 3385534661270087178, 92801587312544823804, 2677687796221222845802, 81124824998424994578652
Offset: 0
The a(4) = 22 sequences:
(1,2,1,2) (2,3,1,3)
(1,2,1,3) (2,3,1,4)
(1,3,1,2) (2,4,1,3)
(1,3,2,3) (3,1,2,1)
(1,3,2,4) (3,1,3,2)
(1,4,2,3) (3,1,4,2)
(2,1,2,1) (3,2,3,1)
(2,1,3,1) (3,2,4,1)
(2,1,3,2) (3,4,1,2)
(2,1,4,3) (4,1,3,2)
(2,3,1,2) (4,2,3,1)
Not requiring non-co-unimodality gives
A328509.
Not requiring non-unimodality also gives
A328509.
The version for run-lengths of partitions is
A332640.
The version for unsorted prime signature is
A332643.
The version for compositions is
A332870.
Unimodal sequences covering an initial interval are
A007052.
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.
Numbers whose negated prime signature is not unimodal are
A332642.
Compositions whose run-lengths are not unimodal are
A332727.
Non-unimodal compositions covering an initial interval are
A332743.
Cf.
A000225,
A000670,
A060223,
A072704,
A329398,
A332281,
A332284,
A332577,
A332578,
A332639,
A332672,
A332834.
-
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[#]&&!unimodQ[-#]&]],{n,0,5}]
-
seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - (1 - 6*x + 12*x^2 - 6*x^3)/((1 - x)*(1 - 2*x)*(1 - 4*x + 2*x^2)), -(n+1)) \\ Andrew Howroyd, Jan 28 2024
A333148
Number of compositions of n whose non-adjacent parts are weakly decreasing.
Original entry on oeis.org
1, 1, 2, 4, 7, 12, 19, 30, 46, 69, 102, 149, 214, 304, 428, 596, 823, 1127, 1532, 2068, 2774, 3697, 4900, 6460, 8474, 11061, 14375, 18600, 23970, 30770, 39354, 50153, 63702, 80646, 101783, 128076, 160701, 201076, 250933, 312346, 387832, 480409, 593716, 732105, 900810, 1106063, 1355336, 1657517, 2023207, 2464987, 2997834, 3639464
Offset: 0
The a(1) = 1 through a(6) = 19 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(111) (31) (32) (33)
(121) (41) (42)
(211) (131) (51)
(1111) (212) (141)
(221) (222)
(311) (231)
(1211) (312)
(2111) (321)
(11111) (411)
(1311)
(2121)
(2211)
(3111)
(12111)
(21111)
(111111)
For example, (2,3,1,2) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), all of which are weakly decreasing.
The case of normal sequences appears to be
A028859.
A version for ordered set partitions is
A332872.
The case of strict compositions is
A333150.
The version for strictly decreasing parts is
A333193.
Standard composition numbers (
A066099) of these compositions are
A334966.
Cf.
A056242,
A059204,
A072706,
A107429,
A115981,
A329398,
A332578,
A332669,
A332673,
A332724,
A332834.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,,y_,_}/;y>x]&]],{n,0,15}]
-
def a333148(n): return number_of_partitions(n) + sum( Partitions(m, max_part=l, length=k).cardinality() * Partitions(n-m-l^2, min_length=k+2*l).cardinality() for l in range(1, (n+1).isqrt()) for m in range((n-l^2-2*l)*l//(l+1)+1) for k in range(ceil(m/l), min(m,n-m-l^2-2*l)+1) ) # Max Alekseyev, Oct 31 2024
A333192
Number of compositions of n with strictly increasing run-lengths.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 14, 16, 24, 31, 37, 51, 67, 76, 103, 129, 158, 199, 242, 293, 370, 450, 538, 652, 799, 953, 1147, 1376, 1635, 1956, 2322, 2757, 3271, 3845, 4539, 5336, 6282, 7366, 8589, 10046, 11735, 13647, 15858, 18442, 21354, 24716, 28630, 32985
Offset: 0
The a(1) = 1 through a(8) = 14 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (122) (33) (133) (44)
(211) (311) (222) (322) (233)
(1111) (2111) (411) (511) (422)
(11111) (3111) (1222) (611)
(21111) (4111) (2222)
(111111) (22111) (5111)
(31111) (11222)
(211111) (41111)
(1111111) (122111)
(221111)
(311111)
(2111111)
(11111111)
For example, the composition (1,2,2,1,1,1) has run-lengths (1,2,3), so is counted under a(8).
Strictly increasing compositions are
A000009.
Partitions with strictly increasing run-lengths are
A100471.
Partitions with strictly decreasing run-lengths are
A100881.
Compositions with equal run-lengths are
A329738.
Compositions whose run-lengths are unimodal are
A332726.
Compositions with strictly increasing or decreasing run-lengths are
A333191.
Numbers with strictly increasing prime multiplicities are
A334965.
Cf.
A072706,
A098859,
A100882,
A100883,
A304686,
A329744,
A329766,
A332726,
A332833,
A332834,
A332835,
A333147,
A333149,
A333190.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Length/@Split[#]&]],{n,0,15}]
b[n_, lst_, v_] := b[n, lst, v] = If[n == 0, 1, If[n <= lst, 0, Sum[If[k == v, 0, b[n - k pz, pz, k]], {pz, lst + 1, n}, {k, Floor[n/pz]}]]]; a[n_] := b[n, 0, 0]; a /@ Range[0, 50] (* Giovanni Resta, May 18 2020 *)
A333193
Number of compositions of n whose non-adjacent parts are strictly decreasing.
Original entry on oeis.org
1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 71, 93, 122, 158, 204, 260, 332, 419, 528, 661, 825, 1023, 1267, 1560, 1916, 2344, 2860, 3476, 4217, 5097, 6147, 7393, 8872, 10618, 12685, 15115, 17977, 21336, 25276, 29882, 35271, 41551, 48872, 57385, 67277, 78745, 92040
Offset: 0
The a(1) = 1 through a(7) = 15 compositions:
(1) (2) (3) (4) (5) (6) (7)
(11) (12) (13) (14) (15) (16)
(21) (22) (23) (24) (25)
(31) (32) (33) (34)
(211) (41) (42) (43)
(221) (51) (52)
(311) (231) (61)
(312) (241)
(321) (322)
(411) (331)
(2211) (412)
(421)
(511)
(2311)
(3211)
For example, (2,3,1,2) is not such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), not all of which are strictly decreasing, while (2,4,1,1) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,1), (4,1), all of which are strictly decreasing.
A version for ordered set partitions is
A332872.
The case of strict compositions is
A333150.
The case of normal sequences appears to be
A001045.
Partitions with strictly increasing run-lengths are
A100471.
Partitions with strictly decreasing run-lengths are
A100881.
Compositions with weakly decreasing non-adjacent parts are
A333148.
Compositions with strictly increasing run-lengths are
A333192.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,,y_,_}/;y>=x]&]],{n,0,15}]
-
\\ p is all, q is those ending in an unreversed singleton.
seq(n)={my(q=O(x*x^n), p=1+q); for(k=1, n, [p,q] = [p*(1 + x^k + x^(2*k)) + q*x^k, q + p*x^k] ); Vec(p)} \\ Andrew Howroyd, Apr 17 2021
A337481
Number of compositions of n that are neither strictly increasing nor strictly decreasing.
Original entry on oeis.org
0, 0, 1, 1, 5, 11, 25, 55, 117, 241, 493, 1001, 2019, 4061, 8149, 16331, 32705, 65461, 130981, 262037, 524161, 1048425, 2096975, 4194097, 8388365, 16776933, 33554103, 67108481, 134217285, 268434945, 536870321, 1073741145, 2147482869, 4294966401, 8589933569
Offset: 0
The a(2) = 1 through a(5) = 11 compositions:
(11) (111) (22) (113)
(112) (122)
(121) (131)
(211) (212)
(1111) (221)
(311)
(1112)
(1121)
(1211)
(2111)
(11111)
A337482 is the semi-strict version.
A337484 counts only compositions of length 3.
A218004 counts strictly increasing or weakly decreasing compositions.
-
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!Greater@@#&]],{n,0,15}]
A242771
Number of integer points in a certain quadrilateral scaled by a factor of n (another version).
Original entry on oeis.org
0, 0, 1, 3, 6, 9, 14, 19, 25, 32, 40, 48, 58, 68, 79, 91, 104, 117, 132, 147, 163, 180, 198, 216, 236, 256, 277, 299, 322, 345, 370, 395, 421, 448, 476, 504, 534, 564, 595, 627, 660, 693, 728, 763, 799, 836, 874, 912, 952, 992, 1033, 1075, 1118, 1161, 1206
Offset: 1
G.f. = x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 19*x^8 + 25*x^9 + 32*x^10 + ...
- E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184).
- E. Ehrhart, Sur un problème de géométrie diophantienne linéaire I, (Polyèdres et réseaux), J. Reine Angew. Math. 226 1967 1-29. MR0213320 (35 #4184). [Annotated scanned copy of pages 16 and 22 only]
- E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II. Systemes diophantiens lineaires, J. Reine Angew. Math. 227 1967 25-49. [Annotated scanned copy of pages 47-49 only]
- Wikipedia, Ehrhart polynomial
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
-
[Floor((5*n-7)*(n-1)/12): n in [1..60]]; // Vincenzo Librandi, Jun 27 2015
-
a[ n_] := Quotient[ 7 - 12 n + 5 n^2, 12];
a[ n_] := With[ {o = Boole[ 0 < n], c = Boole[ 0 >= n], m = Abs@n}, Length @ FindInstance[ 0 < c + x && 0 < c + y && (2 x < c + m && 4 x + 3 y < o + 3 m || m < o + 2 x && 2 x + 3 y < c + 2 m), {x, y}, Integers, 10^9]];
LinearRecurrence[{1,1,0,-1,-1,1},{0,0,1,3,6,9},90] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],!Less@@#&]],{n,0,15}] (* Gus Wiseman, Oct 18 2020 *)
-
{a(n) = (7 - 12*n + 5*n^2) \ 12};
-
{a(n) = if( n<0, polcoeff( x * (2 + x^2 + x^3 + x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^-n), -n), polcoeff( x^3 * (1 + x + x^2 + 2*x^4) / ((1 - x)^2 * (1 - x^6)) + x * O(x^n), n))};
A334966
Numbers k such that the k-th composition in standard order (row k of A066099) has weakly decreasing non-adjacent parts.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 49, 51, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87
Offset: 1
The sequence together with the corresponding compositions begins:
0: () 17: (4,1) 37: (3,2,1)
1: (1) 18: (3,2) 38: (3,1,2)
2: (2) 19: (3,1,1) 39: (3,1,1,1)
3: (1,1) 20: (2,3) 40: (2,4)
4: (3) 21: (2,2,1) 41: (2,3,1)
5: (2,1) 22: (2,1,2) 42: (2,2,2)
6: (1,2) 23: (2,1,1,1) 43: (2,2,1,1)
7: (1,1,1) 24: (1,4) 45: (2,1,2,1)
8: (4) 25: (1,3,1) 47: (2,1,1,1,1)
9: (3,1) 27: (1,2,1,1) 48: (1,5)
10: (2,2) 31: (1,1,1,1,1) 49: (1,4,1)
11: (2,1,1) 32: (6) 51: (1,3,1,1)
12: (1,3) 33: (5,1) 55: (1,2,1,1,1)
13: (1,2,1) 34: (4,2) 63: (1,1,1,1,1,1)
15: (1,1,1,1) 35: (4,1,1) 64: (7)
16: (5) 36: (3,3) 65: (6,1)
For example, (2,3,1,2) is such a composition because the non-adjacent pairs are (2,1), (2,2), (3,2), all of which are weakly decreasing, so 166 is in the sequence
The case of normal sequences appears to be
A028859.
A version for ordered set partitions is
A332872.
These compositions are enumerated by
A333148.
The strict case is enumerated by
A333150.
-
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!MatchQ[stc[#],{_,x_,,y_,_}/;y>x]&]
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