A240077 -id:A240077 - cp's OEIS frontend

A117995 Number of partitions of n in which both smallest and largest part occur only once.

0, 0, 1, 1, 2, 3, 4, 6, 8, 11, 14, 20, 24, 33, 41, 54, 66, 87, 105, 136, 165, 209, 253, 319, 383, 477, 574, 707, 847, 1038, 1238, 1506, 1794, 2166, 2573, 3093, 3660, 4377, 5170, 6152, 7245, 8590, 10087, 11913, 13959, 16423, 19196, 22518, 26252, 30700, 35717

Offset: 1

Emeric Deutsch, Apr 08 2006

nonn

Also number of partitions of n in which the least part is 1 and if k is the largest part, then k>=2 and k-1 also occurs. Example: a(8)=6 because we have [4,3,1],[3,2,2,1],[3,2,1,1,1],[2,2,2,1,1],[2,2,1,1,1,1] and [2,1,1,1,1,1,1].

a(n+1) is the number of partitions of n such that m(greatest part) > m(1), where m = multiplicity, for n>= 0. For example, a(8) counts these 6 partitions of 7: 7, 52, 43, 331, 322, 2221. - Clark Kimberling, Apr 01 2014

			a(8)=6 because we have [7,1],[6,2],[5,3],[5,2,1],[4,3,1] and [3,2,2,1].

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A002865, A240076.

g:=x^3/(1-x)/(1-x^2)+sum(x^(2*k)/product(1-x^j,j=1..k),k=3..70): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..55);

(* See A240077. - Clark Kimberling, Apr 01 2014 *)
sl1Q[n_]:=With[{c=Split[n]},Length[c]>1&&Length[c[[1]]]==Length[c[[-1]]==1]]; Table[Count[IntegerPartitions[n],?(sl1Q)],{n,3,60}] (* _Harvey P. Dale, Apr 06 2025 *)

G.f.: Sum_{k>=2} Sum_{j=1..k-1} x^(j+k)/Product_{i=j+1..k-1} (1-x^i).
G.f.: x^3/[(1-x)(1-x^2)] + Sum_{k>=3} x^(2k)/Product_{j=1..k} (1-x^j).
a(n+1) + A240078(n) = A240080(n) for n >= 0. - Clark Kimberling, Apr 01 2014
a(n) = A002865(n) - (n + 1) mod 2. - Seiichi Manyama, Jan 28 2022

A240076 Number of partitions of n such that m(greatest part) < m(1), where m = multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 8, 13, 18, 27, 35, 52, 67, 93, 121, 164, 209, 279, 353, 461, 582, 748, 935, 1191, 1480, 1861, 2302, 2870, 3526, 4365, 5335, 6554, 7976, 9736, 11789, 14316, 17259, 20844, 25032, 30092, 35992, 43086, 51347, 61215, 72710, 86361, 102235

Offset: 0

Clark Kimberling, Apr 01 2014

			a(7) counts these 6 partitions:  511, 4111, 3211, 31111, 22111, 211111.

Cf. A240077, A240078, A117995, A240080.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, Max[p]] < Count[p, 1]], {n, 0, z}]  (* A240076 *)
t2 = Table[Count[f[n], p_ /; Count[p, Max[p]] <= Count[p, 1]], {n, 0, z}] (* A240077 *)
t3 = Table[Count[f[n], p_ /; Count[p, Max[p]] == Count[p, 1]], {n, 0, z}] (* A240078 *)
t4 = Table[Count[f[n], p_ /; Count[p, Max[p]] > Count[p, 1]], {n, 0, z}] (* A117995 *)
t5 = Table[Count[f[n], p_ /; Count[p, Max[p]] >= Count[p, 1]], {n, 0, z}] (* A240080 *)

a(n) + A240078(n) + A240080(n) = A000041 for n >= 0.

A240078 Number of partitions of n such that m(greatest part) = m(1), where m = multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 3, 6, 6, 10, 9, 18, 16, 27, 29, 44, 46, 71, 75, 109, 122, 167, 188, 257, 290, 382, 442, 569, 657, 840, 971, 1220, 1423, 1761, 2054, 2528, 2944, 3586, 4189, 5061, 5901, 7095, 8262, 9869, 11496, 13652, 15875, 18786, 21805, 25685, 29790

Offset: 0

Clark Kimberling, Apr 01 2014

			a(7) counts these 3 partitions:  61, 421, 1111111.

Cf. A240076, A240077, A117995, A240080.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, Max[p]] < Count[p, 1]], {n, 0, z}]  (* A240076 *)
t2 = Table[Count[f[n], p_ /; Count[p, Max[p]] <= Count[p, 1]], {n, 0, z}] (* A240077 *)
t3 = Table[Count[f[n], p_ /; Count[p, Max[p]] == Count[p, 1]], {n, 0, z}] (* A240078 *)
t4 = Table[Count[f[n], p_ /; Count[p, Max[p]] > Count[p, 1]], {n, 0, z}] (* A117995 *)
t5 = Table[Count[f[n], p_ /; Count[p, Max[p]] >= Count[p, 1]], {n, 0, z}] (* A240080 *)

A240076(n) + a(n) + A240079(n) = A000041(n) for n >= 0.

A240080 Number of partitions of n such that m(greatest part) >= m(1), where m = multiplicity.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 9, 14, 17, 24, 29, 42, 49, 68, 83, 110, 133, 176, 211, 274, 331, 420, 507, 640, 767, 956, 1149, 1416, 1695, 2078, 2477, 3014, 3589, 4334, 5147, 6188, 7321, 8756, 10341, 12306, 14491, 17182, 20175, 23828, 27919, 32848, 38393, 45038, 52505

Offset: 0

Clark Kimberling, Apr 01 2014

			a(7) counts these 9 partitions:  7, 61, 52, 43, 421, 331, 322, 2221, 1111111.

Cf. A240076, A240077, A240078, A117995.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, Max[p]] < Count[p, 1]], {n, 0, z}]  (* A240076 *)
t2 = Table[Count[f[n], p_ /; Count[p, Max[p]] <= Count[p, 1]], {n, 0, z}] (* A240077 *)
t3 = Table[Count[f[n], p_ /; Count[p, Max[p]] == Count[p, 1]], {n, 0, z}] (* A240078 *)
t4 = Table[Count[f[n], p_ /; Count[p, Max[p]] > Count[p, 1]], {n, 0, z}] (* A117995 *)
t5 = Table[Count[f[n], p_ /; Count[p, Max[p]] >= Count[p, 1]], {n, 0, z}] (* A240080 *)

a(n) = A240078(n) + A117995(n) for n >= 0.

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A117995 Number of partitions of n in which both smallest and largest part occur only once.

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A240076 Number of partitions of n such that m(greatest part) < m(1), where m = multiplicity.

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A240078 Number of partitions of n such that m(greatest part) = m(1), where m = multiplicity.

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A240080 Number of partitions of n such that m(greatest part) >= m(1), where m = multiplicity.

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