A309058 -id:A309058 - cp's OEIS frontend

A265250 Number of partitions of n having no parts strictly between the smallest and the largest part (n>=1).

1, 2, 3, 5, 7, 10, 13, 17, 20, 26, 29, 35, 39, 48, 48, 60, 61, 74, 73, 87, 86, 106, 99, 120, 112, 140, 130, 155, 143, 176, 159, 194, 180, 216, 186, 240, 209, 258, 234, 274, 243, 308, 261, 325, 289, 348, 297, 383, 314, 392, 356, 423, 355, 460, 372, 468, 422

Offset: 1

Emeric Deutsch, Dec 25 2015

			a(3) = 3 because we have [3], [1,2], [1,1,1] (all partitions of 3).
a(6) = 10 because we have all A000041(6) = 11 partitions of 6 except [1,2,3].
a(7) = 13 because we have all A000041(7) = 15 partitions of 7 except [1,2,4] and [1,1,2,3].

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.

Cf. A000041, A265249.

Cf. A000005, A002133, A116608, A309058.

g := add(x^i/(1-x^i), i = 1 .. 80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)), j = i+1..80),i=1..80): gser := series(g,x=0,60): seq(coeff(gser,x,n),n=1..50);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
       add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
    end:
a:= n-> b(n$2, 2):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 01 2016

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, If[t == 1, If[Mod[n, i] == 0, 1, 0] + b[n, i - 1, t], Sum[b[n - i*j, i - 1, t - If[j == 0, 0, 1]], {j, 0, n/i}]]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

a(n) = A265249(n,0).
G.f.: G(x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/ ((1-x^i)*(1-x^j)).
a(n) = A116608(n,1) + A116608(n,2) = A000005(n) + A002133(n). - Seiichi Manyama, Sep 14 2023

A309097 Number of partitions of n avoiding the partition (4,2,1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 32, 39, 49, 56, 68, 79, 91, 103, 119, 132, 150, 165, 183, 202, 224, 241, 264, 287, 311, 334, 362, 385, 415, 442, 472, 503, 535, 563, 599, 634, 670, 703, 743, 778, 820, 859, 899, 942, 988, 1027, 1074, 1119, 1167, 1214, 1266

Offset: 0

Jonathan S. Bloom, Jul 12 2019

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.
Conjecture: for n > 0, a(n) is the number of ordered pairs (r, l) such that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l. Actually, such a matrix exists if and only if ceiling(n/(n-r)) <= l <= r+1, see my proof below. If this conjecture is true, then a(n) = (n^2 + 3n)/2 - A006590(n) for n > 0. - Jianing Song, Nov 04 2019
This formula is correct: see links. - Kevin Ryde, Sep 05 2025

Kevin Ryde, Table of n, a(n) for n = 0..6000
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, arXiv:1808.04221 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, arXiv:1808.04238 [math.CO], 2018.
J. Bloom and D. Saracino Rook and Wilf equivalence of integer partitions, European J. Combin., 76 (2018), 199-207.
J. Bloom and D. Saracino On Criteria for rook equivalence of Ferrers boards, European J. Combin., 71 (2018), 246-267.
Jianing Song, Proof that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l if and only if ceiling(n/(n-r)) <= l <= r+1
Kevin Ryde, Proof of formula with ceilings
Wikipedia, Nilpotent matrix

Cf. A006590.

Cf. A309098, A309099, A309058.

lista(n)=my(b(k)=x^k/(1-x^k)+O(x*x^n));Vec(1+sum(i=1,n,b(i)*(1+sum(j=i+1,n,b(j)*(1+b(j+1)))))) \\ Christian Sievers, Sep 01 2025

a(n) = if(n==0,1, n--; my(r,s=sqrtint(n,&r)); n*(n+1)/2 + 1 - r - sum(i=2,s, n\i)<<1); \\ Kevin Ryde, Sep 05 2025

G.f.: 1 + Sum_{i>=1} b(i) * ( 1 + Sum_{j>i} b(j) * ( 1 + b(j+1) ) ) where b(k)=x^k/(1-x^k). - Christian Sievers, Sep 01 2025

More terms from Alois P. Heinz, Jul 12 2019

A309099 Number of partitions of n avoiding the partition (4,3,1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 37, 46, 59, 72, 87, 104, 124, 144, 168, 192, 220, 250, 282, 314, 352, 391, 432, 475, 522, 569, 622, 675, 732, 791, 852, 915, 985, 1055, 1127, 1201, 1281, 1361, 1447, 1533, 1623, 1717, 1813, 1909, 2013, 2118, 2227, 2338, 2453

Offset: 0

Jonathan S. Bloom, Jul 12 2019

nonn

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, arXiv:1808.04221 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, arXiv:1808.04238 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, European J. Combin., 71 (2018), 246-267.
J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, European J. Combin., 76 (2018), 199-207.
Joachim König, Closed-form for the number of partitions of n avoiding the partition (4,3,1), answer to question on MathOverflow (2023).

Cf. A002378, A078567, A309097, A309098, A309058.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[tau](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]+`if`(n=0, 1, n*(1-n)):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 20 2023

b[n_] := b[n] = If[n < 1, {0, 0}, With[{p = b[n-1]},
   p + {DivisorSigma[0, n], p[[1]]}]];
a[n_] := b[n+1][[2]] + If[n == 0, 1, n*(1-n)];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jan 29 2025, after Alois P. Heinz *)

a(n) = if(n == 0, 1, sum(i = 1, n, (n - i + 1) * numdiv(i)) - n * (n - 1)) \\ Mikhail Kurkov, Dec 20 2023 [verification needed]

a(n) = A078567(n+1) - A002378(n-1) for n > 0 with a(0) = 1. - Mikhail Kurkov, Dec 20 2023 [verification needed]

More terms from Alois P. Heinz, Jul 12 2019

A309098 Number of partitions of n avoiding the partition (4,3).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 33, 39, 51, 58, 72, 82, 99, 110, 131, 143, 168, 183, 210, 226, 259, 277, 312, 333, 372, 394, 439, 462, 511, 537, 588, 617, 675, 705, 765, 798, 864, 898, 970, 1005, 1081, 1121, 1199, 1240, 1326, 1369, 1459, 1505, 1599, 1646

Offset: 0

Jonathan S. Bloom, Jul 12 2019

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, arXiv:1808.04221 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, arXiv:1808.04238 [math.CO], 2018.
J. Bloom and D. Saracino, Rook and Wilf equivalence of integer partitions, European J. Combin., 71 (2018), 246-267.
J. Bloom and D. Saracino, On Criteria for rook equivalence of Ferrers boards, European J. Combin., 76 (2018), 199-207.

Cf. A309097, A309099, A309058.

lista(n)=Vec((1+sum(i=3,n,x^i/(1-x^i)+O(x*x^n)))/(1-x-x^2+x^3)) \\ Christian Sievers, Sep 01 2025

G.f.: (1 + Sum_{i>=3} x^i/(1-x^i)) / (1-x-x^2+x^3). - Christian Sievers, Sep 01 2025

More terms from Alois P. Heinz, Jul 12 2019

A364809 Number of partitions of n with at most five part sizes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 791, 1000, 1250, 1565, 1938, 2400, 2945, 3615, 4395, 5342, 6439, 7755, 9268, 11069, 13127, 15537, 18286, 21484, 25095, 29275, 33968, 39344, 45362, 52193, 59836, 68441, 78014, 88724, 100622, 113828

Offset: 0

Seiichi Manyama, Sep 14 2023

nonn

Cf. A265250, A309058, A364793.

Cf. A116608, A365631.

from sympy.utilities.iterables import partitions
def A364809(n): return sum(1 for p in partitions(n) if len(p)<=5) # Chai Wah Wu, Sep 14 2023

a(n) = Sum_{k=1..5} A116608(n,k).

A364793 Number of partitions of n with at most four part sizes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 175, 229, 292, 375, 470, 591, 733, 905, 1103, 1343, 1615, 1938, 2309, 2726, 3211, 3758, 4379, 5069, 5865, 6716, 7694, 8769, 9967, 11254, 12732, 14264, 16025, 17877, 19959, 22149, 24605, 27147, 30012, 33006, 36294, 39742, 43573, 47524

Offset: 0

Seiichi Manyama, Sep 14 2023

nonn

Cf. A265250, A309058, A364809.

Cf. A116608, A365630.

from sympy.utilities.iterables import partitions
def A364793(n): return sum(1 for p in partitions(n) if len(p)<=4) # Chai Wah Wu, Sep 14 2023

a(n) = Sum_{k=1..4} A116608(n,k).

A309194 Number of partitions of n avoiding the partition (4,3,2).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 51, 67, 82, 105, 125, 154, 180, 218, 250, 295, 334, 390, 436, 502, 553, 630, 694, 780, 849, 950, 1027, 1138, 1230, 1355, 1447, 1590, 1694, 1846, 1971, 2133, 2257, 2445, 2579, 2776, 2932, 3142, 3298, 3539, 3702, 3941, 4139

Offset: 0

Jonathan S. Bloom, Jul 16 2019

nonn

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
J. Bloom and D. Saracino Rook and Wilf equivalence of integer partitions, European J. Combin., 76 (2018), 199-207.
J. Bloom and D. Saracino On Criteria for rook equivalence of Ferrers boards, European J. Combin., 71 (2018), 246-267.

Cf. A309097, A309098, A309099, A309058.

More terms from Alois P. Heinz, Jul 18 2019

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A265250 Number of partitions of n having no parts strictly between the smallest and the largest part (n>=1).

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A309097 Number of partitions of n avoiding the partition (4,2,1).

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A309099 Number of partitions of n avoiding the partition (4,3,1).

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A309098 Number of partitions of n avoiding the partition (4,3).

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A364809 Number of partitions of n with at most five part sizes.

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A364793 Number of partitions of n with at most four part sizes.

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A309194 Number of partitions of n avoiding the partition (4,3,2).

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