A343941
Number of strict integer partitions of 2n with reverse-alternating sum 4.
Original entry on oeis.org
0, 0, 1, 0, 1, 2, 3, 3, 4, 5, 7, 8, 10, 11, 14, 15, 18, 20, 23, 25, 29, 31, 35, 38, 42, 45, 50, 53, 58, 62, 67, 71, 77, 81, 87, 92, 98, 103, 110, 115, 122, 128, 135, 141, 149, 155, 163, 170, 178, 185, 194, 201, 210, 218, 227, 235, 245, 253, 263, 272, 282, 291, 302
Offset: 0
The a(2) = 1 through a(12) = 10 strict partitions (empty column indicated by dot, A..D = 10..13):
4 . 521 532 543 653 763 873 983 A93 BA3
631 642 752 862 972 A82 B92 CA2
741 851 961 A71 B81 C91 DA1
64321 65421 65432 76432 76542
75321 75431 76531 86541
76421 86431 87432
86321 87421 87531
97321 97431
98421
A8321
The a(2) = 1 through a(8) = 5 partitions covering an initial interval:
1111 . 32111 33211 33321 333221 543211 543321
322111 332211 3322211 3332221 5432211
3222111 32222111 33222211 33322221
322222111 332222211
3222222111
The non-reverse non-strict version is
A000710.
The non-reverse version is
A026810.
The non-strict version is column k = 2 of
A344610.
A000041 counts partitions of 2n with alternating sum 0, ranked by
A000290.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative:
A344741).
A124754 gives alternating sums of standard compositions (reverse:
A344618).
A316524 is the alternating sum of the prime indices of n (reverse:
A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf.
A000070,
A000097,
A003242,
A006330,
A027187,
A119899,
A152146,
A239830,
A325535,
A344604,
A344607,
A344608,
A344650,
A344739.
-
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==4&]],{n,0,30,2}]
A345961
Numbers whose prime indices have reverse-alternating sum 2.
Original entry on oeis.org
3, 10, 12, 21, 27, 30, 40, 48, 55, 70, 75, 84, 90, 91, 108, 120, 147, 154, 160, 187, 189, 192, 210, 220, 243, 247, 250, 270, 280, 286, 300, 336, 360, 363, 364, 391, 432, 442, 462, 480, 490, 495, 507, 525, 551, 588, 616, 630, 640, 646, 675, 713, 748, 750, 756
Offset: 1
The initial terms and their prime indices:
3: {2}
10: {1,3}
12: {1,1,2}
21: {2,4}
27: {2,2,2}
30: {1,2,3}
40: {1,1,1,3}
48: {1,1,1,1,2}
55: {3,5}
70: {1,3,4}
75: {2,3,3}
84: {1,1,2,4}
90: {1,2,2,3}
91: {4,6}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
Below we use k to indicate reverse-alternating sum.
These multisets are counted by
A000097.
These are the positions of 2's in
A344616.
A000070 counts partitions with alternating sum 1.
A027187 counts partitions with reverse-alternating sum <= 0.
A088218 also counts compositions with alternating sum 0, ranked by
A344619.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A316524 gives the alternating sum of prime indices.
A344606 counts alternating permutations of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf.
A000984,
A001791,
A025047,
A027193,
A239830,
A341446,
A344611,
A344650,
A344651,
A344743,
A345910,
A345911,
A345918.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==2&]
A347465
Numbers whose multiset of prime indices has alternating product > 1.
Original entry on oeis.org
3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127
Offset: 1
The terms and their prime indices begin:
3: {2} 37: {12} 68: {1,1,7}
5: {3} 41: {13} 70: {1,3,4}
7: {4} 42: {1,2,4} 71: {20}
11: {5} 43: {14} 73: {21}
12: {1,1,2} 44: {1,1,5} 75: {2,3,3}
13: {6} 45: {2,2,3} 76: {1,1,8}
17: {7} 47: {15} 78: {1,2,6}
19: {8} 48: {1,1,1,1,2} 79: {22}
20: {1,1,3} 52: {1,1,6} 80: {1,1,1,1,3}
23: {9} 53: {16} 83: {23}
27: {2,2,2} 59: {17} 89: {24}
28: {1,1,4} 61: {18} 92: {1,1,9}
29: {10} 63: {2,2,4} 97: {25}
30: {1,2,3} 66: {1,2,5} 99: {2,2,5}
31: {11} 67: {19} 101: {26}
The squarefree case is
A030059 without 2.
The opposite version (< 1 instead of > 1) is
A119899.
The weak version (>= 1 instead of > 1) is
A344609.
Allowing any integer reverse-alternating product gives
A347454.
Allowing any integer alternating product gives
A347457.
A347446 counts partitions with integer alternating product, reverse
A347445.
Cf.
A008549,
A344607,
A344608,
A344611,
A347442,
A347444,
A347447,
A347453,
A347456,
A347461,
A347462.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],altprod[primeMS[#]]>1&]
A304620
Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 15, 22, 34, 48, 70, 97, 137, 186, 255, 341, 459, 605, 800, 1042, 1359, 1751, 2256, 2879, 3672, 4645, 5869, 7367, 9234, 11508, 14319, 17730, 21916, 26975, 33143, 40570, 49575, 60376, 73402, 88974, 107666, 129933, 156546, 188148, 225767, 270300, 323115, 385453
Offset: 0
The version for even instead of odd greatest part is
A306145.
A000041 counts partitions of 2n with alternating sum 0, ranked by
A000290.
A000070 counts partitions with alternating sum 1.
A067661 counts strict partitions of even length.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf.
A000097,
A006330,
A027193,
A030229,
A067659,
A236559,
A236914,
A239829,
A239830,
A318156,
A338907,
A344611.
-
nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k)/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 + EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 26 2021 *)
A306145
Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k+1) / Product_{j=1..2*k+1} (1 - x^j).
Original entry on oeis.org
0, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 135, 187, 253, 343, 456, 607, 797, 1045, 1355, 1755, 2252, 2884, 3666, 4651, 5863, 7375, 9226, 11517, 14310, 17741, 21904, 26988, 33130, 40586, 49558, 60394, 73383, 88996, 107642, 129958, 156519, 188178, 225734, 270335, 323078, 385494
Offset: 0
The ordered version appears to be
A087447 modulo initial terms.
The version for odd instead of even-length partitions is
A304620.
The case of strict partitions is
A318156.
A000041 counts partitions of 2n with alternating sum 0, ranked by
A000290.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf.
A000070,
A000097,
A006330,
A030229,
A067659,
A236559,
A236914,
A239829,
A239830,
A338907,
A344611.
-
nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k + 1)/Product[(1 - x^j), {j, 1, 2 k + 1}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 23 2021 *)
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 *)
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}]
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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