A265250 -id:A265250 - cp's OEIS frontend

A309058 Partitions of n with parts having at most 3 distinct magnitudes.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 72, 91, 115, 145, 177, 215, 258, 308, 364, 424, 491, 568, 651, 742, 838, 940, 1065, 1181, 1320, 1454, 1619, 1757, 1957, 2124, 2329, 2510, 2763, 2934, 3244, 3432, 3752, 3964, 4329, 4531, 4965, 5179, 5627, 5872, 6391, 6577, 7178, 7405

Offset: 0

Nathan McNew, Jul 09 2019

nonn

Partitions whose Ferrers diagrams do not contain the pattern 4321 under removal of rows and columns (as defined by Bloom and Saracino).

			a(10) = 41 because all of the 42 integer partitions of 10 count (i.e., 10 = 10, 10 = 9+1 = 8+1+1, etc.), except the partition 10 = 4+3+2+1.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
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.

Cf. A265250 (partitions of n with parts having at most 2 distinct magnitudes). Sum of A002134, A002133 and A000005.

Cf. A116608.

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, 3):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2019

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, 3];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz *)

G.f.: Sum_{i>=1} x^i/(1-x^i) + Sum_{j=1..i-1} x^(i+j)/((1-x^i)*(1-x^j)) + Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)*(1-x^k)).

a(n) = Sum_{k=0..3} A116608(n,k). - Alois P. Heinz, Jul 11 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).

A265249 Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 1, 13, 2, 17, 4, 1, 20, 8, 2, 26, 11, 4, 1, 29, 17, 8, 2, 35, 24, 13, 4, 1, 39, 33, 19, 8, 2, 48, 39, 30, 13, 4, 1, 48, 56, 41, 21, 8, 2, 60, 64, 57, 32, 13, 4, 1, 61, 83, 75, 47, 21, 8, 2, 74, 94, 100, 65, 34, 13, 4, 1

Offset: 1

Emeric Deutsch, Dec 25 2015

Number of entries in row n is floor((n-4)/2) (n>=4).
Sum of entries of row n = A000041(n) = number of partitions of n.
T(n,0) = A265250(n).
Sum(k*T(n,k), k>=0) = A182977(n).

			T(8,2) = 1 because among the 22 partitions of 8 only [3,2,2,1] has 2 parts strictly between the smallest and the largest part.
Triangle starts:
1;
2;
3;
5;
7;
10, 1;
13, 2;

CF. A000041, A182977, A265250.

g := add(x^i/(1-x^i), i=1..80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)*mul(1-t*x^k, k=i+1..j-1)),j=i+1..80),i=1..80): gser := simplify(series(g,x=0,23)): for n to 22 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 22 do seq(coeff(P[n],t,k), k=0..degree(P[n])) end do; # yields sequence in triangular form

G.f.: G(t,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)/Product_{k=i+1..j-1} (1-tx^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).

cp's OEIS Frontend

A309058 Partitions of n with parts having at most 3 distinct magnitudes.

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

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A265249 Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).

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

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