A309098 -id:A309098 - cp's OEIS frontend

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

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

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
Wikipedia, Nilpotent matrix

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

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

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

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

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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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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