A237832 -id:A237832 - cp's OEIS frontend

A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

1, 1, 2, 2, 4, 6, 8, 11, 15, 20, 30, 38, 49, 65, 83, 108, 139, 178, 224, 286, 358, 437, 550, 684, 837, 1037, 1269, 1553, 1889, 2295, 2770, 3359, 4035, 4843, 5808, 6951, 8312, 9902, 11752, 13958, 16531, 19541, 23037, 27162, 31911, 37488, 43950, 51463, 60127, 70229

Offset: 0

Gus Wiseman, Mar 17 2023

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (22)   (32)    (33)    (43)
                   (31)   (41)    (42)    (52)
                   (211)  (221)   (51)    (61)
                          (311)   (321)   (322)
                          (2111)  (411)   (331)
                                  (2211)  (421)
                                  (3111)  (511)
                                          (2221)
                                          (3211)
                                          (4111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The complement is counted by A360254, ranks A360558.
These partitions have ranks A361395.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, reverse A058398.
A067538 counts partitions with integer mean, strict A102627.
A116608 counts partitions by number of distinct parts.
Cf. A106529, A144300, A237363, A237832, A240219, A316413, A349156.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),
     `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2023

Table[Length[Select[IntegerPartitions[n],2*Length[Union[#]]>=Length[#]&]],{n,0,30}]

A237831 Number of partitions of n such that (greatest part) - (least part) <= number of parts.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 18, 23, 32, 40, 57, 70, 94, 120, 157, 196, 256, 318, 408, 508, 640, 792, 996, 1223, 1518, 1863, 2296, 2798, 3432, 4162, 5070, 6130, 7422, 8936, 10777, 12916, 15500, 18522, 22136, 26348, 31376, 37222, 44160, 52236, 61756, 72824, 85847

Offset: 1

Clark Kimberling, Feb 16 2014

			a(6) = 10 counts all the 11 partitions of 6 except 4+1+2.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..93 from R. J. Mathar)
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Cf. A109083, A237830, A237832, A237833, A237834.

z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*k*(x^(k*(3*k-1)/2)+x^(k*(3*k+1)/2)))) \\ Seiichi Manyama, May 20 2023

a(n) + A237833(n) = A000041(n). - R. J. Mathar, Nov 24 2017

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * k * ( x^(k*(3*k-1)/2) + x^(k*(3*k+1)/2) ). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023

A237833 Number of partitions of n such that (greatest part) - (least part) > number of parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 20, 31, 41, 56, 74, 101, 129, 172, 219, 284, 362, 463, 579, 735, 918, 1147, 1422, 1767, 2172, 2680, 3279, 4013, 4888, 5947, 7200, 8721, 10515, 12663, 15202, 18235, 21798, 26039, 31015, 36898, 43802, 51930, 61426, 72590

Offset: 1

Clark Kimberling, Feb 16 2014

			a(8) = 4 counts these partitions:  7+1, 6+2, 6+1+1, 5+2+1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..96 from R. J. Mathar)
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Cf. A237830, A237831, A237832, A237834.

Different from, but has the same beginning as, A275633.

isA237833 := proc(p)
    if abs(p[1]-p[-1]) > nops(p) then
        return 1;
    else
        return 0;
    end if;
end proc:
A237833 := proc(n)
    local a,p;
    a := 0 ;
    p := combinat[firstpart](n) ;
    while true do
        a := a+isA237833(p) ;
        if nops(p) = 1 then
            break;
        end if;
        p := nextpart(p) ;
    end do:
    return a;
end proc:
seq(A237833(n),n=1..20) ; # R. J. Mathar, Nov 17 2017

z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)

my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^k*(k-1)*(x^(k*(3*k-1)/2)+x^(k*(3*k+1)/2))))) \\ Seiichi Manyama, May 20 2023

A237831(n) + a(n) = A000041(n). - R. J. Mathar, Nov 24 2017

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^k * (k-1) * ( x^(k*(3*k-1)/2) + x^(k*(3*k+1)/2) ). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023

A318176 a(n) is the number of integer partitions of n for which the greatest part minus the least part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 2, 3, 2, 4, 8, 4, 15, 12, 16, 21, 29, 30, 48, 40, 74, 67, 105, 102, 148, 154, 210, 223, 285, 292, 437, 428, 593, 630, 842, 894, 1168, 1317, 1628, 1759, 2249, 2426, 3112, 3356, 4158, 4637, 5647, 6172, 7657, 8400, 10146, 11401, 13450, 15069, 17948, 20108, 23674, 26867, 31398, 35133

Offset: 1

Nick Mayers, Aug 20 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n)>0 for n=1,2 and n>4. To see this: for n=1,2 take the partitions (1) and (1,1), respectively; for n>3 odd take the partition (2,...,2,1,1,1); for n>2 congruent to 2 (mod 6), say n=6k+2, take the partition (2k+1,2k,2k,1); for n>4 congruent to 4 (mod 6), say n=6k+4, take the partition (2k+1,k+1,k+1,k+1,k); for n>0 congruent to 0 (mod 6), say n=6k, take the partition (2k,1,...,1) with 4k ones.

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A318177, A318178, A237832, A318196, A318203

A318177 a(n) is the number of integer partitions of n for which the Kimberling index is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 2, 3, 2, 2, 5, 5, 8, 8, 11, 18, 20, 26, 26, 35, 49, 56, 73, 88, 101, 130, 148, 182, 207, 260, 310, 385, 455, 579, 657, 800, 910, 1135, 1310, 1546, 1763, 2169, 2488, 2936, 3352, 3962, 4612, 5435, 6187, 7370, 8430, 9951, 11276, 13236, 15133, 17624, 20009, 23551, 26464

Offset: 1

Nick Mayers and Melissa Mayers, Aug 20 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n)>0 for n=4 and n>5. To see this: for n>0 congruent to 0 (mod 4), say 4k+4, take the partition of the form (2k+3,2k+1); for n congruent to 2 (mod 4) if n=6 take (4,4,1), if n=10 take (5,3,2), if n>10, say n=4k+10, take the partition (2k+7,2k-1,1,1,1,1); for n>1 congruent to 1 (mod 6), say n=6k+1, take the partition (2k+3,2k-1,2k-1); for n>5 congruent to 5 (mod 6), say n=6k+5, take the partition (2k+3,2k+3,2k-1); for n>3 congruent to 3 (mod 6), say n=6k-3, take the partition (2k+1,2,...,2) with 2k-2 2's.

George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.
V. Coll, M. Hyatt, C. Magnant, and H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev and A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A318176, A318178, A237832, A318196, A318203

A318178 a(n) is the number of integer partitions of n for which the length is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 1, 2, 1, 8, 9, 5, 8, 15, 10, 17, 21, 24, 25, 45, 43, 68, 53, 82, 81, 143, 111, 165, 168, 247, 232, 314, 313, 442, 491, 587, 596, 918, 842, 1217, 1304, 1645, 1650, 2221, 2311, 2922, 3119, 4007, 4184, 5521, 5699, 7232, 7498, 9543, 9580, 12802

Offset: 1

Nick Mayers, Aug 20 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n)>0 for n=2,6 and n>8. To see this: for n congruent to 2,6 (mod 8) take the partition of the form (2,...,2); for n>=9 congruent to 1,5 (mod 8), say n=4k+1, take the partition (4k-3,3,1); for n>7 congruent to 3 (mod 8), say n=8k+3, take the partition (4k,3,2,...,2) with 2k 2's; for n>7 congruent to 7 (mod 8) take the partition ((n-1)/2, (n-5)/2,3); for n>8 congruent to 4 (mod 8) take the partition (n-8,4,3,1); and for n>8 congruent to 0 (mod 8) take the partition (n-8,4,4).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A318176, A318177, A237832, A318196, A318203

A318196 a(n) is the number of integer partitions of n for which the smallest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 4, 7, 7, 16, 11, 23, 22, 43, 41, 61, 56, 97, 103, 126, 146, 205, 210, 274, 315, 389, 461, 531, 623, 751, 901, 968, 1227, 1372, 1661, 1787, 2238, 2332, 2998, 3105, 3921, 4103, 5241, 5148, 6778, 6795, 8745, 8683, 11231, 11133, 14523, 14246, 18284, 18121, 23536, 22790, 29627, 29143, 36990

Offset: 1

Nick Mayers, Aug 20 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n)>0 for n>2. To see this: for n odd, say n=2k+3, take the partition (2k+1,1,1); for n even, say n=2k+4, take the partition (2k+1,1,1,1).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A237832, A318176, A318177, A318178, A318203

A237830 Number of partitions of n such that (greatest part) - (least part) < number of parts.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 47, 62, 81, 105, 135, 174, 222, 282, 357, 450, 565, 707, 880, 1093, 1353, 1669, 2052, 2517, 3077, 3753, 4565, 5539, 6704, 8097, 9755, 11730, 14075, 16854, 20142, 24029, 28611, 34009, 40355, 47807, 56542, 66772, 78728

Offset: 1

Clark Kimberling, Feb 16 2014

			a(6) = 8 counts these partitions:  6, 3+3, 4+1+1, 3+2+1, 2+2+2, 3+1+1+1, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..95 from R. J. Mathar)
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Cf. A000070, A237831-A237834.

z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*x/(1-x)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k-1)/2)*(1-x^(2*k)))) \\ Seiichi Manyama, May 20 2023

a(n) + A237834(n) = A000041(n). - R. J. Mathar, Nov 24 2017

G.f.: (1/Product_{k>=1} (1-x^k)) * (x/(1-x)) * Sum_{k>=1} (-1)^(k-1) * x^(3*k*(k-1)/2) * (1-x^(2*k)). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023

A237834 Number of partitions of n such that (greatest part) - (least part) >= number of parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 10, 15, 20, 30, 39, 54, 71, 96, 123, 163, 208, 270, 342, 437, 548, 695, 865, 1083, 1341, 1666, 2048, 2527, 3089, 3784, 4604, 5606, 6786, 8222, 9907, 11940, 14331, 17196, 20554, 24563, 29252, 34820, 41327, 49016, 57982, 68545, 80833

Offset: 1

Clark Kimberling, Feb 16 2014

			a(7) = 4 counts these partitions:  6+1, 5+2, 5+1+1, 4+2+1.

R. J. Mathar, Table of n, a(n) for n = 1..95
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Cf. A237830, A237831, A237832, A237833.

z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]>=Length[#]&)],{n,50}] (* _Harvey P. Dale, Jul 21 2023 *)

A237830(n)+a(n) = A000041(n). - R. J. Mathar, Nov 24 2017

A318203 a(n) is the number of integer partitions of n for which the largest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 3, 3, 3, 5, 6, 6, 10, 13, 18, 18, 27, 31, 40, 52, 58, 78, 95, 103, 136, 161, 194, 225, 265, 346, 386, 483, 585, 660, 797, 938, 1134, 1316, 1521, 1832, 2081, 2550, 2901, 3407, 3913, 4614, 5345, 6305, 7280, 8514, 9824, 11377, 13120, 14960, 17427, 19981, 23316, 26859, 30390

Offset: 1

Nick Mayers, Melissa Mayers Aug 21 2018

nonn

The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([,]) were [,] denotes the bracket multiplication on g.
For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
a(n) > 0 for n = 3, 4 and n > 5. To see this: for n odd if n=3 take the partition (1,1,1), if n > 5 take the partition (2,...,2,1,1,1,1,1); for n > 2 congruent to 2 (mod 6), say n=6k+2, take the partition (2k,1,...,1) with 4k+2 1's; for n > 0 congruent to 4 (mod 6), say n=6k+4, take the partition (2k+1, 1,...,1) with 4k+3 1's; for n > 0 congruent to 0 (mod 6), say n=6k, take the partition (2k, 2k, 2k-1, 1).

V. Coll, M. Hyatt, C. Magnant, H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
V. Dergachev, A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.

Cf. A237832, A318176, A318177, A318178, A318196

cp's OEIS Frontend

A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

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A237831 Number of partitions of n such that (greatest part) - (least part) <= number of parts.

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A237833 Number of partitions of n such that (greatest part) - (least part) > number of parts.

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A318176 a(n) is the number of integer partitions of n for which the greatest part minus the least part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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A318177 a(n) is the number of integer partitions of n for which the Kimberling index is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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A318178 a(n) is the number of integer partitions of n for which the length is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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A318196 a(n) is the number of integer partitions of n for which the smallest part is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

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A237830 Number of partitions of n such that (greatest part) - (least part) < number of parts.

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A237834 Number of partitions of n such that (greatest part) - (least part) >= number of parts.

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