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Search: a240178 -id:a240178
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A240179
Number of partitions of n such that (least part) <= (multiplicity of greatest part).
+10
6
0, 1, 1, 2, 4, 5, 8, 11, 17, 23, 33, 43, 61, 79, 108, 140, 187, 238, 314, 397, 513, 648, 826, 1032, 1307, 1622, 2029, 2508, 3113, 3821, 4713, 5754, 7048, 8569, 10431, 12618, 15290, 18413, 22193, 26628, 31954, 38184, 45639, 54340, 64694, 76780, 91077, 107732
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also, for n >= 0, a(n) is the number of partitions of n such that (greatest part) >= (multiplicity of least part). For n >=1, a(n) is also the number of partitions of n such that (least part) >= (multiplicity of greatest part), as well as the number of partitions p of n such that min(p) = min(c(p)), where c = conjugate..
LINKS
Table of n, a(n) for n=0..47.
FORMULA
a(n) = A240178(n) + A240180(n), for n >= 0.
2*a(n) + A240180(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 8 partitions: 51, 411, 321, 3111, 222, 2211, 21111, 111111.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}] (* A240178 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}] (* A240178, n>0 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)
CROSSREFS
Cf. A240178, A240180, A000041.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 02 2014
STATUS
approved
A240057
Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).
+10
3
0, 2, 3, 4, 6, 10, 14, 21, 28, 40, 53, 74, 97, 131, 171, 225, 290, 377, 480, 616, 779, 987, 1238, 1556, 1935, 2411, 2981, 3685, 4527, 5562, 6793, 8295, 10081, 12241, 14805, 17890, 21538, 25906, 31062, 37201, 44429, 53004, 63070, 74964, 88898, 105297
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Let # denote "number of" and c(p) = conjugate of partitionp. Then
A240057(n) = # p such that min(p) not = max(c(p));
A039899(n) = # p such that min(p) < max(c(p));
A039900(n) = # p such that min(p) <= max(c(p));
A006141(n) = # p such that min(p) = max(c(p));
A003114(n) = # p such that min(p) > max(c(p));
A003016(n) = # p such that min(p) >= max(c(p));
A064173(n) = # p such that max(p) < max(c(p));
A064174(n) = # p such that max(p) <= max(c(p));
A047993(n) = # p such that max(p) = max(c(p)).
See A240178 for related sequences. - Clark Kimberling, Apr 11 2014
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
FORMULA
a(n) + A006141(n) = A000041(n) for n > 0.
EXAMPLE
a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.
MAPLE
b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
end:
a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..50); # Alois P. Heinz, Apr 03 2014
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}] (* A003106 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)
tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]];
a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)
CROSSREFS
Cf. A003106, A003114, A006141, A039899, A039900.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 02 2014
STATUS
approved
A240180
Number of partitions of n such that (least part) = (multiplicity of greatest part).
+10
3
0, 1, 0, 1, 3, 3, 5, 7, 12, 16, 24, 30, 45, 57, 81, 104, 143, 179, 243, 304, 399, 504, 650, 809, 1039, 1286, 1622, 2006, 2508, 3077, 3822, 4666, 5747, 6995, 8552, 10353, 12603, 15189, 18371, 22071, 26570, 31785, 38104, 45419, 54213, 64426, 76596, 90710
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Also the number of partitions p of n such that min(p) = min(conjugate(p)). Example:a(7) counts these 7 partitions: 61, 511, 421, 4111, 3211, 31111, 211111, of which the respective conjugates are 211111, 31111, 3211, 4111, 421, 511, 61. - Clark Kimberling, Apr 11 2014
LINKS
Table of n, a(n) for n=0..47.
FORMULA
a(n) = A240179(n) - A240178(n), for n >= 0.
a(n) + 2*A240178(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 5 partitions: 51, 411, 321, 3111, 21111.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}] (* A240178 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}] (* A240178, n>0 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)
CROSSREFS
Cf. A240178, A240179, A000041.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 02 2014
STATUS
approved
A240182
Number of partitions of n such that (greatest part) <= (multiplicity of least part).
+10
3
1, 1, 1, 1, 3, 2, 5, 4, 8, 10, 13, 15, 25, 25, 37, 46, 61, 70, 97, 112, 150, 177, 224, 270, 347, 407, 508, 611, 754, 895, 1106, 1304, 1594, 1892, 2283, 2708, 3262, 3835, 4595, 5421, 6452, 7574, 8993, 10530, 12445, 14564, 17123, 19992, 23465, 27302, 31931
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..50.
FORMULA
a(n) = A240178(n) + A240183(n), for n >= 1.
a(n) + A240179(n) = A000041(n) for n >= 0.
EXAMPLE
a(8) counts these 8 partitions: 41111, 32111, 311111, 2222, 22211, 221111, 2111111, 11111111.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Min[p]]], {n, 0, z}] (* A240178 except for n=0 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240182 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Min[p]]], {n, 0, z}] (* A240183 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Min[p]]], {n, 0, z}] (* A240184 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240179 *)
CROSSREFS
Cf. A240178, A240183, A240184, A240179, A000041.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 02 2014
STATUS
approved
A240183
Number of partitions of n such that (greatest part) = (multiplicity of least part).
+10
3
0, 1, 0, 0, 2, 0, 2, 0, 3, 3, 4, 2, 9, 3, 10, 10, 17, 11, 26, 19, 36, 33, 48, 47, 79, 71, 101, 109, 149, 151, 215, 216, 293, 318, 404, 443, 575, 611, 773, 864, 1068, 1175, 1458, 1609, 1964, 2210, 2642, 2970, 3577, 3995, 4753, 5369, 6332, 7138, 8414, 9476
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..55.
FORMULA
A240178(n) + a(n) + A240184(n) = A000041(n) for n >= 0.
EXAMPLE
a(8) counts these 3 partitions: 41111, 32111, 22211.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Min[p]]], {n, 0, z}] (* A240178 except for n=0 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240182 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Min[p]]], {n, 0, z}] (* A240183 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Min[p]]], {n, 0, z}] (* A240184 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240179 *)
CROSSREFS
Cf. A240178, A240182, A240184, A240179, A000041.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 02 2014
STATUS
approved
A240184
Number of partitions of n such that (greatest part) > (multiplicity of least part).
+10
3
0, 0, 1, 2, 2, 5, 6, 11, 14, 20, 29, 41, 52, 76, 98, 130, 170, 227, 288, 378, 477, 615, 778, 985, 1228, 1551, 1928, 2399, 2964, 3670, 4498, 5538, 6755, 8251, 10027, 12175, 14715, 17802, 21420, 25764, 30886, 37009, 44181, 52731, 62730, 74570, 88435, 104762
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Table of n, a(n) for n=0..47.
FORMULA
A240178(n) + A240183(n) + a(n ) = A000041(n) for n >= 1.
EXAMPLE
a(6) counts these 6 partitions: 6, 51, 42, 411, 33, 321.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Min[p]]], {n, 0, z}] (* A240178 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240182 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Min[p]]], {n, 0, z}] (* A240183 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Min[p]]], {n, 0, z}] (* A240184 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240179 *)
CROSSREFS
Cf. A240178, A240182, A240183, A240179, A000041.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 02 2014
STATUS
approved
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