A351593
Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318
Offset: 0
The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
1 2 3 4 5 6 7 8 9 A B C D E F
221 331 332 441 442 443 552 553 554 663
551 661 662 771
33221 44221 44331
55221
With only equalities we get:
- opposite odd-length:
A000009 (except at 0)
Without equalities we get:
- opposite any length:
A122129 (apparently)
- opposite odd-length:
A122130 (apparently)
- even-length:
A122134 (apparently)
-
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]
A318156
Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 55, 69, 85, 104, 127, 154, 186, 224, 268, 320, 381, 452, 534, 630, 741, 869, 1017, 1187, 1382, 1606, 1862, 2155, 2489, 2869, 3301, 3792, 4349, 4979, 5692, 6497, 7405, 8429, 9581, 10876, 12331, 13963, 15792, 17840, 20131, 22691
Offset: 0
From _Gus Wiseman_, Jul 18 2021: (Start)
Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
(2,1) (3,2) (4,3) (5,4) (6,5) (7,6) (8,7) (9,8)
(4,1) (5,2) (6,3) (7,4) (8,5) (9,6) (10,7)
(6,1) (7,2) (8,3) (9,4) (10,5) (11,6)
(8,1) (9,2) (10,3) (11,4) (12,5)
(10,1) (11,2) (12,3) (13,4)
(12,1) (13,2) (14,3)
(6,4,2,1) (14,1) (15,2)
(6,4,3,2) (16,1)
(8,4,2,1) (6,5,4,2)
(8,4,3,2)
(8,6,2,1)
(10,4,2,1)
Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
(End)
The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
- Allowing any length gives
A036469.
- The non-strict version is
A306145.
- Allowing any number of odd parts gives
A343942 (odd bisection of
A067661).
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
Cf.
A000070,
A030229,
A035294,
A058696,
A078616,
A087447,
A152146,
A236559,
A343941,
A344611,
A344739.
-
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);
-
nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] - QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 18 2021 *)
A351594
Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.
Original entry on oeis.org
0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863
Offset: 0
The a(1) = 1 through a(9) = 7 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(111) (221) (222) (331) (332) (333)
(11111) (22111) (441)
(1111111) (22221)
(33111)
(2211111)
(111111111)
The ordered version (compositions) is
A016116 shifted right once.
All odd-length partitions are counted by
A027193.
Replacing equal with unequal relations appears to give:
The case that is also strict at even indices is:
-
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]
A346634
Number of strict odd-length integer partitions of 2n + 1.
Original entry on oeis.org
1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937
Offset: 0
The a(0) = 1 through a(7) = 14 partitions:
(1) (3) (5) (7) (9) (11) (13) (15)
(4,2,1) (4,3,2) (5,4,2) (6,4,3) (6,5,4)
(5,3,1) (6,3,2) (6,5,2) (7,5,3)
(6,2,1) (6,4,1) (7,4,2) (7,6,2)
(7,3,1) (7,5,1) (8,4,3)
(8,2,1) (8,3,2) (8,5,2)
(8,4,1) (8,6,1)
(9,3,1) (9,4,2)
(10,2,1) (9,5,1)
(10,3,2)
(10,4,1)
(11,3,1)
(12,2,1)
(5,4,3,2,1)
The even version is the even bisection of
A067661.
The case of all odd parts is counted by
A069911 (non-strict:
A078408).
A340385 counts partitions with odd length and maximum, ranked by
A340386.
Other cases of odd length:
-
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Aug 05 2021
-
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]
A349149
Number of even-length integer partitions of n with at most one odd part in the conjugate partition.
Original entry on oeis.org
1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 7, 12, 11, 19, 15, 30, 22, 45, 30, 67, 42, 97, 56, 139, 77, 195, 101, 272, 135, 373, 176, 508, 231, 684, 297, 915, 385, 1212, 490, 1597, 627, 2087, 792, 2714, 1002, 3506, 1255, 4508, 1575, 5763, 1958, 7338, 2436, 9296, 3010, 11732
Offset: 0
The a(2) = 1 through a(9) = 7 partitions:
11 21 22 32 33 43 44 54
1111 2111 2211 2221 2222 3222
111111 3211 3311 3321
211111 221111 4311
11111111 222111
321111
21111111
The case of 0 odd conjugate parts is
A000041 up to 0's, ranked by
A000290.
The case of 1 odd conjugate part is
A000070 up to 0's.
A045931 counts partitions with as many even as odd parts, ranked by
A325698.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A122111 is a representation of partition conjugation.
A277103 counts partitions with the same alternating sum as their conjugate.
A277579 counts partitions with as many even parts as odd conjugate parts.
A325039 counts partitions with the same product as their conjugate.
A344610 counts partitions by sum and positive reverse-alternating sum.
A345196 counts partitions with the same rev-alt sum as their conjugate.
Cf.
A000097,
A000700,
A001700,
A027187,
A027193,
A108711,
A236559,
A236913,
A325534,
A344607,
A344651.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[conj[#],_?OddQ]<=1&]],{n,0,30}]
A361656
Number of odd-length integer partitions of n with integer mean.
Original entry on oeis.org
0, 1, 1, 2, 1, 2, 4, 2, 1, 9, 8, 2, 13, 2, 16, 51, 1, 2, 58, 2, 85, 144, 57, 2, 49, 194, 102, 381, 437, 2, 629, 2, 1, 956, 298, 2043, 1954, 2, 491, 2293, 1116, 2, 4479, 2, 6752, 14671, 1256, 2, 193, 8035, 4570, 11614, 22143, 2, 28585, 39810, 16476, 24691, 4566
Offset: 0
The a(1) = 1 through a(10) = 8 partitions (A = 10):
1 2 3 4 5 6 7 8 9 A
111 11111 222 1111111 333 22222
321 432 32221
411 441 33211
522 42211
531 43111
621 52111
711 61111
111111111
For example, the partition (3,3,2,1,1) has length 5 and mean 2, so is counted under a(10).
The even-length version is counted by
A361655.
A326622 counts factorizations with integer mean.
-
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,30}]
-
a(n)=if(n==0, 0, sumdiv(n, d, if(d%2, polcoef(1/prod(k=1, d, 1 - x^k + O(x^(n-d+1))), n-d)))) \\ Andrew Howroyd, Mar 24 2023
A318155
Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k*(2*k+1)) / Product_{j=1..2*k} (1 - x^j).
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 44, 55, 68, 84, 103, 126, 153, 185, 223, 268, 320, 381, 452, 535, 631, 742, 870, 1018, 1188, 1383, 1607, 1863, 2155, 2489, 2869, 3301, 3792, 4348, 4978, 5691, 6496, 7404, 8428, 9580, 10875, 12330, 13962, 15791, 17840, 20131, 22691
Offset: 0
A000070 counts partitions with alternating sum 1.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf.
A027193,
A035294,
A067659,
A087447,
A236559,
A236914,
A239829,
A306145,
A344611,
A344739,
A346634.
-
nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k + 1))/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] + QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 29 2021 *)
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