A238406 -id:A238406 - cp's OEIS frontend

A238394 Number of partitions of n that sorted in increasing order do not contain a part k in position k.

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 8, 9, 12, 13, 17, 22, 28, 34, 42, 48, 59, 71, 88, 106, 130, 151, 181, 210, 250, 295, 354, 417, 494, 577, 675, 780, 909, 1053, 1231, 1431, 1668, 1930, 2240, 2573, 2963, 3392, 3896, 4461, 5129, 5873, 6742, 7710, 8816, 10043, 11439

Offset: 0

Giovanni Resta, Feb 26 2014

nonn

The definition forbids partitions with a part equal to 1, so the smallest possible part is 2, which however can appear at most once.

Note that considering partitions in standard decreasing order, we obtain A064428.

			a(6) = 3, because of the 11 partitions of 6 only 3 do not contain a 1 in position 1, a 2 in position 2, or a 3 in position 3, namely (3,3), (2,4) and (6).
From _Joerg Arndt_, Mar 23 2014: (Start)
There are a(15) = 22 such partitions of 15:
01:  [ 2 3 4 6 ]
02:  [ 2 3 5 5 ]
03:  [ 2 3 10 ]
04:  [ 2 4 4 5 ]
05:  [ 2 4 9 ]
06:  [ 2 5 8 ]
07:  [ 2 6 7 ]
08:  [ 2 13 ]
09:  [ 3 3 4 5 ]
10:  [ 3 3 9 ]
11:  [ 3 4 8 ]
12:  [ 3 5 7 ]
13:  [ 3 6 6 ]
14:  [ 3 12 ]
15:  [ 4 4 7 ]
16:  [ 4 5 6 ]
17:  [ 4 11 ]
18:  [ 5 5 5 ]
19:  [ 5 10 ]
20:  [ 6 9 ]
21:  [ 7 8 ]
22:  [ 15 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Giovanni Resta)

Cf. A000041, A238395, A064428, A001522, A238352.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
       x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 26 2014

a[n_] := Length@ Select[ IntegerPartitions@n, 0 < Min@ Abs[ Reverse@# - Range@ Length@#] &]; Array[a, 30]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]]; a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]} ] ][b[n, n]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)

a(n) + A238395(n) = p(n) = A000041(n).
a(n) = Sum_{k>=0} A238406(n,k). - Alois P. Heinz, Feb 26 2014
a(n) = A238352(n,0). - Alois P. Heinz, Jun 08 2014

A244239 Number of partitions of n into 3 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330, 340

Offset: 9

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Column k=3 of A238406.

a:= proc(n) option remember; `if`(n<14, [1, 3, 4, 6, 7][n-8],
      ((-4*n+56)*a(n-5) +(3*n-16)*a(n-4) +(7*n-66)*a(n-3)
      +(4*n-44)*a(n-2) +(28-3*n)*a(n-1)) / (7*n-78))
    end:
seq(a(n), n=9..80);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]];
a[n_] := Coefficient[b[n, n], x, 3];
Table[a[n], {n, 9, 80}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)

From Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 15.
G.f.: x^9*(-x^6 + 2*x^4 + x^3 - 2*x - 1)/((x - 1)^3*(x + 1)*(x^2 + x + 1)). (End)

A244240 Number of partitions of n into 4 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 4, 7, 11, 15, 19, 25, 30, 37, 44, 53, 61, 72, 82, 95, 107, 122, 136, 154, 170, 190, 209, 232, 253, 279, 303, 332, 359, 391, 421, 457, 490, 529, 566, 609, 649, 696, 740, 791, 839, 894, 946, 1006, 1062, 1126, 1187, 1256, 1321, 1395, 1465, 1544, 1619, 1703

Offset: 14

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 14..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Column k=4 of A238406.

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {p_q(N+M)/(p_q(M)*p_q(N)); }
A_q(N) = {my(q='q+O('q^N), g=sum(i=3,N, q^(8+i) * (GB_q(3,i) - q^2 - q^3 - sum(j=0,i, q^j))));
Vec(g)}
A_q(70) \\ John Tyler Rascoe, Apr 23 2024

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 26.
G.f.: x^14*(-x^4 + x + 1)*(x^8 - x^5 - 2*x^4 + 2*x + 1)/((x - 1)^4*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)
G.f.: Sum_{i>2} q^(8+i) * ( q_binomial(3,i) - q^2 - q^3 - Sum_{j=0..i} (q^j) ). - John Tyler Rascoe, Apr 23 2024

A244241 Number of partitions of n into 5 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 5, 11, 19, 29, 39, 53, 67, 84, 103, 126, 150, 179, 210, 246, 285, 329, 376, 430, 487, 551, 619, 695, 775, 864, 958, 1061, 1170, 1289, 1414, 1551, 1694, 1849, 2012, 2188, 2372, 2570, 2777, 2999, 3231, 3479, 3737, 4013, 4300, 4605, 4923, 5260, 5610, 5981

Offset: 20

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 20..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Column k=5 of A238406.

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-6) - a(n-7) + a(n-8) + a(n-9) + a(n-10) - a(n-13) - a(n-14) + a(n-15) for n > 40.
G.f.: x^20*(-x^20 + 2*x^16 + 2*x^15 + 4*x^14 + x^13 - 3*x^12 - 7*x^11 - 9*x^10 - 5*x^9 + 2*x^8 + 8*x^7 + 9*x^6 + 8*x^5 + x^4 - 3*x^3 - 5*x^2 - 4*x - 1)/((x - 1)^5*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)). (End)

A244242 Number of partitions of n into 6 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 6, 16, 31, 52, 76, 107, 143, 184, 233, 289, 354, 427, 512, 606, 716, 835, 972, 1122, 1292, 1476, 1685, 1909, 2161, 2432, 2734, 3057, 3417, 3799, 4222, 4673, 5168, 5693, 6270, 6879, 7545, 8249, 9014, 9821, 10698, 11619, 12616, 13665, 14795, 15981, 17259

Offset: 27

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 27..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Column k=6 of A238406.

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) - 2*a(n-7) + a(n-9) + a(n-10) + a(n-11) + a(n-12) - 2*a(n-14) - a(n-16) + a(n-19) + a(n-20) - a(n-21) for n > 57.
G.f.: x^27*(-x^30 + 2*x^25 + 2*x^24 + 2*x^23 + 4*x^22 + 2*x^21 + x^20 - 9*x^19 - 12*x^18 - 16*x^17 - 12*x^16 + x^15 + 13*x^14 + 24*x^13 + 25*x^12 + 20*x^11 + 3*x^10 - 11*x^9 - 23*x^8 - 22*x^7 - 15*x^6 - 6*x^5 + 5*x^4 + 9*x^3 + 9*x^2 + 5*x + 1)/((x - 1)^6*(x + 1)^3*(x^2 + 1)*(x^2 - x + 1)*(x^2 + x + 1)^2*(x^4 + x^3 + x^2 + x + 1)). (End)

A244243 Number of partitions of n into 7 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 7, 22, 48, 88, 140, 207, 291, 389, 508, 646, 809, 995, 1212, 1457, 1742, 2061, 2425, 2833, 3295, 3808, 4386, 5024, 5737, 6522, 7394, 8349, 9406, 10559, 11827, 13208, 14721, 16361, 18153, 20090, 22198, 24472, 26938, 29591, 32462, 35543, 38866, 42427, 46258

Offset: 35

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 35..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Column k=7 of A238406.

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28) for n > 77.
G.f.: x^35*(-x^42 + 2*x^36 + 2*x^35 + 2*x^34 + 2*x^33 + 4*x^32 + x^31 + 2*x^30 - 3*x^29 - 8*x^28 - 14*x^27 - 25*x^26 - 24*x^25 - 16*x^24 + 4*x^23 + 29*x^22 + 50*x^21 + 58*x^20 + 56*x^19 + 28*x^18 - 8*x^17 - 47*x^16 - 75*x^15 - 76*x^14 - 60*x^13 - 28*x^12 + 10*x^11 + 42*x^10 + 55*x^9 + 53*x^8 + 33*x^7 + 14*x^6 - 5*x^5 - 18*x^4 - 19*x^3 - 14*x^2 - 6*x - 1)/(x^28 - x^27 - x^26 + x^23 + x^21 + x^20 - x^18 - x^17 - 2*x^16 + 2*x^12 + x^11 + x^10 - x^8 - x^7 - x^5 + x^2 + x - 1). (End)

A244244 Number of partitions of n into 8 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 8, 29, 71, 142, 245, 384, 567, 792, 1069, 1401, 1796, 2260, 2801, 3427, 4150, 4981, 5925, 7004, 8222, 9602, 11151, 12895, 14838, 17016, 19430, 22120, 25090, 28385, 32004, 36004, 40384, 45203, 50463, 56231, 62506, 69367, 76809, 84919, 93697, 103237, 113534

Offset: 44

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 44..1000

Column k=8 of A238406.

A244245 Number of partitions of n into 9 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 9, 37, 101, 220, 410, 684, 1062, 1553, 2174, 2943, 3875, 4995, 6319, 7876, 9689, 11798, 14219, 17005, 20179, 23798, 27894, 32531, 37743, 43610, 50169, 57509, 65679, 74778, 84858, 96036, 108373, 121997, 136983, 153471, 171546, 191373, 213040, 236731, 262556

Offset: 54

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 54..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Column k=9 of A238406.

A244246 Number of partitions of n into 10 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Original entry on oeis.org

1, 10, 46, 139, 329, 660, 1173, 1918, 2938, 4277, 5990, 8123, 10742, 13903, 17678, 22140, 27383, 33487, 40560, 48710, 58055, 68734, 80881, 94657, 110223, 127771, 147478, 169580, 194278, 221842, 252503, 286579, 324327, 366120, 412247, 463132, 519112, 580667

Offset: 65

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 65..1000
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 5.

Column k=10 of A238406.

cp's OEIS Frontend

A238394 Number of partitions of n that sorted in increasing order do not contain a part k in position k.

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A244239 Number of partitions of n into 3 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244240 Number of partitions of n into 4 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244241 Number of partitions of n into 5 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244242 Number of partitions of n into 6 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244243 Number of partitions of n into 7 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244244 Number of partitions of n into 8 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244245 Number of partitions of n into 9 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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A244246 Number of partitions of n into 10 parts such that every i-th smallest part (counted with multiplicity) is different from i.

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