A076269 -id:A076269 - cp's OEIS frontend

A006463 Convolve natural numbers with characteristic function of triangular numbers.

0, 0, 1, 2, 4, 6, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 45, 50, 55, 60, 65, 70, 76, 82, 88, 94, 100, 106, 112, 119, 126, 133, 140, 147, 154, 161, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 340, 350, 360

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) = length (i.e., number of elements minus 1) of longest chain in partition lattice Par(n). Par(n) is the set of partitions of n under "dominance order": partition P is <= partition Q iff the sum of the largest k parts of P is <= the corresponding sum for Q for all k.
If C_n(q, t) are the (q, t)-Catalan polynomials, then p_n(x) := C_n(x, x) is a polynomial in x such that a(n) is the degree of the lowest degree term. The sequence of polynomials p_n(x) = 1, 1, 2*x, x^2 + 4*x^3, 3*x^4 + 4*x^5 + 7*x^6 + ... while the coefficient of the lowest degree term is A074909(n). - Michael Somos, Jan 09 2019
If f is a strictly convex function computed on partitions of n (A000041), then a(n)+1 provides a lower bound on the number of distinct values of n taken by f across all partitions of n. - Noah A Rosenberg, Apr 18 2025

			a(6)=8; one longest chain consists of these 9 partitions: 6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1. Others are obtained by changing 3+3 to 4+1+1 or 2+2+2 to 3+1+1+1.
G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 14*x^8 + 17*x^9 + ...

N. A. Rosenberg, Mathematical Properties of Population-Genetic Statistics, Princeton University Press, 2025; page 113.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.2(f).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Curtis Greene and Daniel J. Kleitman, Longest chains in the lattice of integer partitions ordered by majorization, Europ. J. Combinator. 7 (1986), 1-10.
Jeffrey Shallit, Letter to N. J. A. Sloane with attachment, Aug. 1979
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A076269, A074909, A010054, A060432.

0 together with the partial sums of A003056.

a006463 n = a006463_list !! n
a006463_list = 0 : scanl1 (+) a003056_list
-- Reinhard Zumkeller, Dec 17 2011

a[n_] := (x = Quotient[ Sqrt[1+8*n]-1, 2]; x*(x^2-1+3*(n-x*(x+1)/2))/3); Table[a[n], {n, 0, 58}] (* Jean-François Alcover, Apr 11 2013, after Michael Somos *)
t = {0}; Do[Do[AppendTo[t, t[[-1]]+n], {k, 0, n}], {n, 0, 11}]; t (* Jean-François Alcover, May 10 2016, after Vladimir Joseph Stephan Orlovsky *)
Join[{0},Table[ListConvolve[Range[x],Table[If[OddQ[Sqrt[8n+1]],1,0],{n,x}]],{x,0,60}]//Flatten] (* Harvey P. Dale, Jan 14 2019 *)

{a(n) = my(x); if( n<0, 0, x = (sqrtint(8*n + 1) - 1)\2; x * (x^2 - 1 + 3 * (n - x*(x+1)/2)) / 3)}; /* Michael Somos, Mar 06 2006 */

from math import isqrt
def A006463(n): return (m:=isqrt((n<<3)+1)-1>>1)*(6*n-2-m*(m+3))//6 # Chai Wah Wu, Jun 07 2025

Let n=binomial(m+1, 2)+r, 0<=r<=m; then a(n) = (1/3)*m*(m^2+3*r-1).
G.f.: (psi(x) - 1) * x / (1 - x)^2 where psi() is a Ramanujan theta function. - Michael Somos, Mar 06 2006
a(n) = Sum_(k=0..n-1) A003056(k). - Daniele Parisse, Jul 10 2007
a(n+1) - 2*a(n) + a(n-1) = A010054(n) if n>0. - Michael Somos, May 07 2016

Edited by Dean Hickerson, Nov 09 2002

A077765 Number of maximum-size antichains in partition lattice Par(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 2, 4, 15, 4, 2, 11, 18, 14, 53, 2, 54, 1606, 482, 104, 754, 536

Offset: 0

Dean Hickerson, Nov 14 2002

Par(n) is the set of partitions of n under 'dominance order': partition P is <= partition Q iff the sum of the largest k parts of P is <= the corresponding sum for Q for all k.

			For n=10, the maximum size is A076269(10)=4. There are 2 maximum-size antichains: {5+1+1+1+1+1, 4+3+1+1+1, 4+2+2+2, 3+3+3+1} and {6+1+1+1+1, 5+2+2+1, 4+4+1+1, 4+3+3}. So a(10)=2.

The corresponding sizes are in A076269.

Mathematica
```
leq[p_, q_] := If[Length[p]
				
```

A337206 Cardinality of maximal level sets of Gini index on integer partitions.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7, 8, 9, 11, 13, 15, 17, 21, 23, 28, 33, 38, 44, 52, 60, 72, 81, 95, 112, 128, 147, 175, 195, 233, 267, 305, 353, 412, 462, 533, 617, 703, 807, 932, 1052, 1210, 1389, 1569, 1785, 2060, 2315, 2642, 3023, 3405, 3876, 4413, 4968

Offset: 0

Grant Kopitzke, Aug 18 2020

nonn

a(n) is a lower bound on A076269(n).

			For n=6 the maximal level set of the Gini index contains the partitions (3,3) and (4,1,1). So a(6)=2.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Grant Kopitzke, The Gini Index of an Integer Partition, arXiv:2005.04284 [math.CO], 2020.

Lower bound on A076269.

Cf. A161680, A264034.

b:= proc(n, i, w) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, w)+expand(x^(w*i)*b(n-i, min(i, n-i), w+1))))
    end:
a:= n-> max(coeffs(b(n$2, 0))):
seq(a(n), n=0..61);  # Alois P. Heinz, Jan 20 2023

m = 75;
p = Product[ 1/(1 - q^Binomial[i + 1, 2] x^i), {i, 1, m}];
psn = Expand@Normal@Series[ p, {x, 0, m}];
psnc = CoefficientList[CoefficientList[psn, {x}, {m}], {q}];
Map[Max, psnc]

G.f.: Product_{n=1..oo} 1/(1-q^(binomial(n+1,2))x^n)-1 = Sum_{n=1..oo} Sum_{lambda a partition of n} q^(binomial(n+1,2)-g(lambda))x^n, where g(lambda) is the Gini index of lambda.

a(n) = max_{k=0..A161680(n)} A264034(n,k). - Alois P. Heinz, Jan 20 2023

Typo in a(43) corrected by Alois P. Heinz, Jan 20 2023

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A006463 Convolve natural numbers with characteristic function of triangular numbers.

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A077765 Number of maximum-size antichains in partition lattice Par(n).

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A337206 Cardinality of maximal level sets of Gini index on integer partitions.

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A076779 Maximum number of disjoint maximum-size antichains in partition lattice Par(n).

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