Jeremy Lovejoy - cp's OEIS frontend

A356367 Number of plane partitions of n having exactly one row and one column, each of equal length.

1, 1, 1, 2, 2, 5, 6, 11, 16, 26, 36, 58, 81, 122, 172, 251, 350, 502, 692, 972, 1332, 1842, 2499, 3414, 4592, 6200, 8277, 11064, 14656, 19424, 25544, 33584, 43880, 57274, 74362, 96429, 124468, 160422, 205942, 263938, 337083, 429768

Offset: 0

Jeremy Lovejoy, Oct 16 2022

nonn

The empty plane partition of 0 contributes an initial term equal to 1.

Also equal to the number of unimodal compositions of n+1 where the peak appears exactly once and the number of parts to the left of the peak is equal to the number of parts to the right of the peak.

			For n = 7 the valid unimodal compositions of n+1=8 are (8), (1,6,1), (1,5,2), (2,5,1), (3,4,1), (1,4,3), (2,4,2), (1,1,4,1,1), (1,1,3,2,1), (1,2,3,1,1) and (1,1,1,2,1,1,1), and so a(7) = 11.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
B. Kim and J. Lovejoy, The rank of a unimodal sequence and a partial theta identity of Ramanujan, Int. J. Number Theory 10 (2014), 1081-1098.

Cf. A006330, A195012.

nmax = 50; CoefficientList[Series[1 + 1/Product[(1 - x^k)^2, {k, 1, nmax}] * Sum[(-1)^(k + r + 1) * x^(k*(k + 1)/2 + r*(r + 1)/2 + 2*k*r)*(1 - x^r), {r, 0, nmax}, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 28 2023 *)

G.f.: 1 + (1/Product_{n>=1}(1-x^n)^2)*Sum_{r,n>=0}(-1)^(n+r+1)*x^(n*(n+1)/2 + r*(r+1)/2 + 2*n*r)*(1-x^r).

A357459 The total number of fixed points among all partitions of n, when parts are written in nondecreasing order.

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 10, 17, 22, 34, 46, 66, 88, 123, 160, 218, 283, 375, 482, 630, 799, 1030, 1299, 1651, 2066, 2602, 3230, 4032, 4976, 6157, 7554, 9288, 11326, 13837, 16793, 20393, 24632, 29763, 35783, 43031, 51527, 61683, 73577, 87729, 104252, 123834, 146664

Offset: 0

Jeremy Lovejoy, Sep 29 2022

nonn

For instance, the partition (1,3,3,3,5) = (y(1),y(2),y(3),y(4),y(5)) has 3 fixed points, since y(i) = i for i=1,3,5.

			The 7 partitions of 5 are (1,1,1,1,1), (1,1,1,2), (1,2,2), (1,1,3), (1,4), (2,3), and (5), containing 1, 1, 2, 2, 1, 0, and 0 fixed points, respectively, and so a(5) = 1+1+2+2+1+0+0=7.

A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

Cf. A001522 (parts decreasing), A099036.

G.f.: (Product_{k>=1}(1/(1-q^k)))*Sum_{n>=1}q^(2*n-1)*Product_{k=n..2*n-2}(1-q^k).

A357320 The total number of fixed points among all strict partitions of n, when parts are written in increasing order.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 4, 3, 4, 4, 9, 8, 11, 12, 15, 21, 24, 28, 34, 40, 46, 60, 67, 80, 93, 110, 125, 148, 174, 200, 231, 268, 306, 354, 404, 461, 534, 606, 690, 786, 895, 1012, 1150, 1298, 1467, 1662, 1872, 2104, 2374, 2664, 2990, 3355, 3759, 4202, 4702, 5256

Offset: 0

Jeremy Lovejoy, Sep 29 2022

nonn

For instance, the partition (1,2,4,7,11) = (y(1),y(2),y(3),y(4),y(5)) has 2 fixed points, since y(1) = 1 and y(2) = 2.

			The 10 strict partition of 10 are (1,2,3,4), (2,3,5), (1,4,5), (1,3,6), (4,6), (1,2,7), (3,7), (2,8), (1,9), and (10), containing 4,0,1,1,0,2,0,0,1, and 0 fixed points, respectively, and so a(10) = 9.

For the same count but where parts are written in decreasing order, see A352829.

For the case of ordinary partitions, see A357459.

G.f.: (Product_{k>=1}(1+q^k))*Sum_{n>=1}q^(n*(n+1)/2)/Product_{k=1..n}(1+q^k).

A347207 The number of overpartitions of n whose Frobenius symbols have only positive parts in the top row.

Original entry on oeis.org

1, 0, 2, 4, 6, 10, 16, 26, 40, 62, 92, 136, 198, 284, 404, 570, 794, 1100, 1512, 2060, 2792, 3760, 5030, 6696, 8868, 11682, 15322, 20008, 26012, 33688, 43464, 55864, 71560, 91360, 116256, 147490, 186562, 235304, 295976, 371308, 464614, 579944, 722180, 897212

Offset: 0

Jeremy Lovejoy, Aug 23 2021

nonn

a(n) is also the excess of the number of overpartitions of n with an even number of overlined parts larger than the number of non-overlined parts over the number of overpartitions of n with an odd number of overlined parts larger than the number of non-overlined parts.

B. Kim, E. Kim, and J. Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int. J. Number Theory 17 (2021), 603-619.

Cf. A015128, A143184, A347206.

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k))*(1-2*Sum_{n>=1} q^(n*(3*n-1)/2)*(1-q^n)).

A347206 The number of overpartitions of n whose Frobenius symbols have only odd parts in the top row.

Original entry on oeis.org

1, 0, 2, 2, 4, 4, 8, 10, 16, 20, 30, 38, 54, 68, 94, 120, 160, 202, 266, 334, 432, 540, 688, 856, 1080, 1334, 1668, 2052, 2542, 3110, 3828, 4660, 5698, 6906, 8394, 10130, 12250, 14720, 17716, 21210, 25412, 30310, 36172, 42994, 51114, 60558, 71740, 84732, 100052

Offset: 0

Jeremy Lovejoy, Aug 23 2021

nonn

a(n) is also the excess of the number of overpartitions of n with an even number of non-overlined parts larger than the number of overlined parts over the number of overpartitions of n with an odd number of non-overlined parts larger than the number of overlined parts.

B. Kim, E. Kim, and J. Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int. J. Number Theory 17 (2021), 603-619.

Cf. A015128, A143184, A347207.

G.f.: (Product_{k>=1} 1/(1-q^k))*Sum_{n>=0} q^(n*(3*n+1)/2)*(1-q^(2*n+1)).

A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 17, 29, 49, 79, 125, 193, 293, 437, 642, 932, 1336, 1896, 2663, 3709, 5121, 7020, 9551, 12913, 17347, 23172, 30779, 40679, 53495, 70030, 91269, 118459, 153133, 197214, 253057, 323595, 412418, 523953, 663612, 838035, 1055304, 1325287, 1659969

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

Also equal to A340658(n) - A001524(n).

			a(4) = 5 counts the overpartitions [3,1], [2,2], [2,1,1], [1,1,1,1], and [1',1,1,1].

B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340658, A340659.

b:= proc(n, i, c) option remember; `if`(n=0,
     `if`(c>1, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
      add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c > 1, 1, 0], If[i < 1, 0, b[n, i-1, c] + Sum[
     Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) - Sum_{n>=0} q^(n*(n+1)/2)/ ((Product_{k=1..n} (1-q^k)^2) * (1-q^(n+1))).

A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.

Original entry on oeis.org

1, 0, 1, 2, 3, 5, 7, 11, 15, 23, 31, 45, 61, 85, 114, 156, 206, 276, 363, 477, 621, 808, 1041, 1339, 1713, 2182, 2769, 3501, 4409, 5534, 6927, 8635, 10741, 13316, 16467, 20303, 24980, 30643, 37518, 45815, 55836, 67889, 82395, 99772, 120609, 145501, 175229, 210637

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

			a(5) = 5 counts the overpartitions [4',1], [4,1'], [3',2], [3,2'], and [2',1',1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340658.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c=0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
       add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

b[n_, i_, c_] := b[n, i, c] = If[n==0, If[c==0, 1, 0], If[i<1, 0, b[n, i-1, c] + Sum[Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[1 + Sum[x^(j*(j+1)/2 + j) / QPochhammer[x, x, j]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2021 *)

G.f.: Sum_{n>=0} q^(n*(n+1)/2 + n)/Product_{k=1..n} (1-q^k)^2.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * phi^2 * n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 06 2021
a(n) = A143184(n) - A001524(n). - Vaclav Kotesovec, Jun 06 2021

a(0)=1 prepended by Alois P. Heinz, Jan 15 2021

A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 25, 41, 67, 105, 163, 246, 368, 540, 784, 1124, 1596, 2242, 3124, 4316, 5918, 8058, 10899, 14651, 19581, 26028, 34417, 45293, 59327, 77372, 100483, 129984, 167502, 215077, 275199, 350966, 446162, 565451, 714515, 900334, 1131370, 1417963

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

			a(3) = 4 counts the overpartitions [3], [2,1], [1,1,1], and [1',1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340659.

b:= proc(n, i, c) option remember; `if`(n=0,
     `if`(c>0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
      add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

b[n_, i_, c_] := b[n, i, c] = If[n==0, If[c>0, 1, 0], If[i<1, 0, b[n, i-1, c] + Sum[Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) - Sum_{n>=0} q^(n*(n+1)/2)/Product_{k=1..n} (1-q^k)^2.

A338860 The excess of the number of partitions of n with more odd parts than even parts over the number of partitions of n with more even parts than odd parts.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 4, 6, 8, 11, 17, 21, 30, 38, 53, 68, 90, 115, 150, 192, 243, 312, 390, 496, 613, 775, 951, 1193, 1456, 1810, 2200, 2715, 3285, 4026, 4856, 5909, 7106, 8595, 10301, 12394, 14809, 17728, 21118, 25171, 29891, 35489, 42018, 49702, 58678, 69180

Offset: 0

Jeremy Lovejoy, Jan 12 2021

nonn

			The 3 partitions of 4 with more odd parts than even parts are [3,1], [2,1,1], and [1,1,1,1], while the 2 partitions of 4 with more even parts than odd parts are [4] and [2,2].   Hence a(4) = 3-2 = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A108949, A108950.

b:= proc(n, i, t) option remember; `if`(n=0, signum(t), `if`(i<1, 0,
      b(n, i-1, t)+ b(n-i, min(n-i, i), t+(2*irem(i, 2)-1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 14 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t], If[i < 1, 0,
   b[n, i-1, t] + b[n-i, Min[n-i, i], t + (2*Mod[i, 2]-1)]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Sep 09 2022, after Alois P. Heinz *)

for(n=0,43,my(me=0,mo=0);forpart(v=n,my(x=Vec(v),se=sum(k=1,#x,x[k]%2==0),so=sum(k=1,#x,x[k]%2>0));me+=(se>so);mo+=(so>se));print1(mo-me,", ")) \\ Hugo Pfoertner, Jan 13 2021

G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2-n)*(1-q^n)/Product_{k=1..n} (1-q^(2*k))^2.

a(n) = A108950(n) - A108949(n).

A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 3, 5, 7, 8, 13, 14, 23, 23, 37, 39, 59, 63, 90, 101, 136, 156, 201, 239, 296, 355, 428, 523, 617, 754, 878, 1078, 1243, 1517, 1741, 2121, 2426, 2928, 3348, 4021, 4596, 5468, 6257, 7400, 8472, 9936, 11389, 13285, 15233, 17645, 20244, 23346

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(7) = 3 counts the partitions [4,2,1], [3,2,2], and [2,2,2,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A340621, A340622, A340647.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c<0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k - 1))/(1 - x^(2*k)), {k, 1, nmax/2}] - Sum[x^(k^2)/Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(prod(k=1, N, (1+x^(2*k-1))/(1-x^(2*k)))-sum(k=0, sqrt(N), x^(k^2)/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

G.f.: (Product_{k>=1} (1+q^(2*k-1))/(1-q^(2*k))) - Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^(2*k))^2.

cp's OEIS Frontend

User: Jeremy Lovejoy

A356367 Number of plane partitions of n having exactly one row and one column, each of equal length.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A357459 The total number of fixed points among all partitions of n, when parts are written in nondecreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A357320 The total number of fixed points among all strict partitions of n, when parts are written in increasing order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A347207 The number of overpartitions of n whose Frobenius symbols have only positive parts in the top row.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A347206 The number of overpartitions of n whose Frobenius symbols have only odd parts in the top row.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A340668 The number of overpartitions of n where the number of non-overlined parts is at least two more than the number of overlined parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A340658 The number of overpartitions of n having more non-overlined parts than overlined parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica