Edward Early - cp's OEIS frontend

User: Edward Early

Edward Early has authored 5 sequences.

A058886 Sum of the row of the character table of S_n corresponding to the partition 2,1^{n-2}.

2, 1, 2, 3, 4, 5, 5, 7, 9, 11, 12, 15, 18, 20, 23, 28, 33, 37, 42, 48, 56, 63, 70, 80, 92, 102, 114, 129, 145, 161, 178, 199, 223, 246, 271, 302, 335, 368, 404, 447, 493, 540, 591, 649, 713, 779, 848, 929, 1017, 1106, 1203, 1312, 1429, 1551, 1682, 1828, 1986

Offset: 2

Edward Early, Jan 08 2001

nonn

Also partial sums of the number of self-conjugate partitions (A000700) with the last term subtracted. - Edward Early, May 30 2012

More terms added by Edward Early, May 30 2012

A076269 Size of largest antichain in partition lattice Par(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 9, 10, 11, 14, 17, 20, 24, 29, 35, 40, 48, 55

Offset: 0

Edward Early, Nov 05 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.

			a(10)=4; one antichain consists of 5+1+1+1+1+1, 4+3+1+1+1, 4+2+2+2 and 3+3+3+1.

T. Brylawski, The lattice of integer partitions, Discrete Math. 6 (1973), 201-219.
Edward Early, Chain Lengths in the Dominance Lattice, June 8, 2013.
Edward Early, Chain Lengths in the Dominance Lattice, Discrete Mathematics, Volume 313, Issue 20, 28 October 2013, Pages 2168-2177.
C. Greene and D. J. Kleitman, Longest Chains in the Lattice of Integer Partitions ordered by Majorization, Europ. J. Combinatorics 7 (1986), 1-10.
Grant Kopitzke, The Gini Index of an Integer Partition, arXiv:2005.04284 [math.CO], 2020. Mentions this sequence.

Cf. A006463, A077765, A076779.

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

Order of growth is between n^(-5/2)e^(Pi*sqrt(2n/3)) and n^(-1)e^(Pi*sqrt(2n/3)).

Edited by Dean Hickerson, Nov 09 2002

a(22)-a(26) by Paul Tabatabai, Dec 05 2018

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

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 2, 3, 7, 3, 2, 3, 4, 3, 3, 2, 5, 5, 2, 1

Offset: 0

Edward Early, Nov 15 2002

Edward Early, Chain Lengths in the Dominance Lattice, FPSAC'03, 15th International Conference on Formal Power Series and Algebraic Combinatorics, Linköping University, (Sweden), June 23-27 2003.
Edward Early, Chain Lengths in the Dominance Lattice, Discrete Mathematics, Volume 313, Issue 20, 28 October 2013, Pages 2168-2177.

Cf. A076269, A077765.

a(17)-a(19) from Sean A. Irvine, Apr 16 2025

A058884 Partial sums of the partition function (A000041), with the last term subtracted. Also the sum of the row of the character table for S_n corresponding to the partition n-1,1 for n>1. Also the sum over all partitions lambda of n of one less than the number of 1's in lambda.

Original entry on oeis.org

-1, 0, 0, 1, 2, 5, 8, 15, 23, 37, 55, 83, 118, 171, 238, 332, 453, 618, 827, 1107, 1460, 1922, 2504, 3253, 4188, 5380, 6860, 8722, 11024, 13895, 17421, 21787, 27122, 33677, 41653, 51390, 63179, 77496, 94755, 115600, 140632, 170725, 206717, 249804, 301151, 362367, 435077, 521439, 623674, 744695

Offset: 0

Edward Early, Jan 08 2001

For n>=1 number of up-steps in all partitions of n (represented as weakly increasing lists), see example. - Joerg Arndt, Sep 03 2014

			a(6) = 8 because the 11 partitions of 6
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 1 3 ]
04:  [ 1 1 2 2 ]
05:  [ 1 1 4 ]
06:  [ 1 2 3 ]
07:  [ 1 5 ]
08:  [ 2 2 2 ]
09:  [ 2 4 ]
10:  [ 3 3 ]
11:  [ 6 ]
contain 0+1+1+1+1+2+1+0+1+0+0 = 8 up-steps. - _Joerg Arndt_, Sep 03 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Anders Claesson, Atli Fannar Franklín, and Einar Steingrímsson, Permutations with few inversions, arXiv:2305.09457 [math.CO], 2023.
S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, Quaestiones Mathematicae 34 (2011), 187-202.

Cf. A000041, A000070.

Cf. A218074 (up-steps in partitions into distinct parts).

a:= proc(n) uses combinat; add(numbpart(k), k=0..n-1)-numbpart(n) end:
seq(a(n), n=0..49);

p[n_] := IntegerPartitions[n]; l[n_] := Length[p[n]]; Table[Count[Flatten[p[n]], 1] - l[n], {n, 0, 30}] (* Clark Kimberling, Mar 08 2012 *)

a(n) = {sum(k=0, n-1, numbpart(k)) - numbpart(n)} \\ Andrew Howroyd, Apr 21 2023

Vec((2*x - 1)/(1 - x)/eta(x + O(x^51))) \\ Andrew Howroyd, Apr 21 2023

From Andrew Howroyd, Apr 21 2023: (Start)
a(n) = A000070(n-1) - A000041(n) for n > 0.
G.f.: (2*x - 1)*P(x)/(1 - x) where P(x) is the g.f. of A000041. (End)

More terms from James Sellers, Sep 28 2001

A045883 a(n) = ((3n+1)2^n - (-1)^n)/9.

Original entry on oeis.org

0, 1, 3, 9, 23, 57, 135, 313, 711, 1593, 3527, 7737, 16839, 36409, 78279, 167481, 356807, 757305, 1601991, 3378745, 7107015, 14913081, 31224263, 65244729, 136081863, 283348537, 589066695, 1222872633, 2535223751, 5249404473, 10856722887, 22429273657, 46290203079

Offset: 0

Edward Early

Without the initial zero, PSumSIGN transform of A001787. - Michael Somos, Jul 10 2003
Number of rises (drops) in the compositions of n+2 with parts in N.
From Michel Lagneau, Jan 13 2012: (Start)
This sequence is connected with the Collatz problem. We consider the array T(i,j) where the i-th row gives the parity trajectory of i, for example for i = 6, the infinite trajectory is 6 -> 3 -> 10 ->5 -> 16 ->8 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1 -> 4->2-> 1 ... and T(6,j) = [0,1,0,1,0,0,0,0,1,0,0,1,...,1,0,0,1,...]. Now, we consider the sum of the digits 1 of each array T(i,j), where
a(1) = sum of the digits "1" of T(i,j), i = 1..2^1 and j = 1;
a(2) = sum of the digits "1" of T(i,j), i = 1..2^2 and j = 1..2;
a(3) = sum of the digits "1" of T(i,j), i = 1..2^3 and j = 1..3;
a(n) = Sum_{i=1..2^n}(Sum_{j=1..n} T(i,j)) = Sum_{i=1..n} A001045(n)*2^(n-i) = convolution of A001045 and A000079 (see the formula below).
The number of digits "0" equals A113861(n) = n*2^n - a(n) because n and 2^n are the dimensions of each array.
An important result is that the ratio r = A113861(n) / A045883(n) tends towards 2 when n tends towards infinity. In other words, when the array tends towards infinity, the ratio r = (number of divisions by 2) / (number of multiplications by 3) tends towards 2, even if there exists divergent trajectories. That is the problem! For each possible divergent infinite trajectory, r < 2 even though the global ratio r is 2.
Conclusion:
1. For each number n with a convergent trajectory T(n,k), k = 1..infinity, or for each row of the array T(i,j), the ratio r tends towards 2 (the proof is easy because the trajectory becomes periodic from a certain index 1001001001...).
2. For each array of dimension n X 2^n, the radio r tends towards 2.
3. If there exists a number n such that the trajectory is divergent, this trajectory is random and r tends towards a real x such that 1 < = r < = x < 2.
4. In order to establish a proof of the Collatz problem from this considerations (if that is possible), it is necessary to prove that a ratio < 2 for an infinite row (or several rows) of an infinite array T(i,j) is incompatible with r = 2, the exact ratio for this array. (End)
a(n) is the distance spectral radius of the dimension-regular generalized recursive circulant graph (commonly known as multiplicative circulant graph) of order 2^n. - John Rafael M. Antalan, Sep 25 2020
Total sum over all compositions of n of the absolute differences between consecutive parts, assuming an initial part 0. - Alois P. Heinz, Apr 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
John Rafael M. Antalan and Francis Joseph H. Campeña, Distance eigenvalues and forwarding indices of dimension-regular generalized recursive circulant graph of order power of two and three, arXiv:2009.11608[math.CO], 2020.
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
Tomislav Došlić and Biserka Kolarec, On Log-Definite Tempered Combinatorial Sequences, Mathematics (2025) Vol. 13, Iss. 7, 1179.
Silvia Heubach and Toufik Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
F. K. Hwang, Three versions of a group testing game, SIAM J. Algebraic Discrete Methods 5 (1984), no. 2, 145--153. MR0745434(85d:90120). See p. 151, f(n) (but divide by 2). - _N. J. A. Sloane_, Apr 13 2014
Peter J. Larcombe and Eric J. Fennessey, On a Scaled Balanced-Power Product Recurrence, Fibonacci Quart. 54 (2016), no. 3, 242-246. See Remark 2.2 p. 244.
Peter J. Larcombe, Julius Fergy T. Rabago, and Eric J. Fennessey, On two derivative sequences from scaled geometric mean sequence terms, Palestine Journal of Mathematics (2018) Vol. 7(2), 397-405.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Partial sums of A059570, bisection: A014916.
Row sums of triangle A094953.
Cf. A059260, A127984.
Cf. A000079, A001045, A001787, A045883, A049260, A113861, A140960, A188545.
Cf. A238343, A238344.

[((3*n+1)*2^n-(-1)^n)/9: n in [0..35]]; // Vincenzo Librandi, Jun 15 2017

A045883:=n->((3*n+1)*2^n-(-1)^n)/9; seq(A045883(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014

nn=31;a=x^2(1-x)/(1-x-2x^2)/(1-2x);b=x^2/(1-2x)^2;Drop[CoefficientList[Series[(b-a)/2,{x,0,nn}],x],2] (* Geoffrey Critzer, Mar 21 2014 *)
CoefficientList[Series[x / ((1 + x) (1 - 2 x)^2), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 15 2017 *)
LinearRecurrence[{3, 0, -4}, {0, 1, 3}, 33] (* Jean-François Alcover, Sep 27 2017 *)

{a(n) = if( n<-1, 0, ((3*n + 1)*2^n - (-1)^n) / 9)};

G.f.: x/((1+x)*(1-2*x)^2).
a(n) = 3*a(n-1) - 4*a(n-3).
Convolution of A001045 and A000079. G.f.: x/((1-2*x)(1-x-2*x^2)). - Paul Barry, May 21 2004
Starting with "1" = triangle A049260 * the odd integers as a vector. - Gary W. Adamson, Mar 06 2012
a(n) = A140960(n)/2. - J. M. Bergot, May 21 2013
From Wolfdieter Lang, Jun 14 2017: (Start)
a(n) = f(n)*2^n, where f(n) is a rational Fibonacci type sequence based on fuse(a,b) = (a+b+1)/2 with f(0) = 0, f(1) = 1/2 and f(n) = fuse(f(n-1), f(n-2)), for n >= 2. For fuse(a,b) see the Jeff Erickson link under A188545. Proof with f(n) = (3*n+1 - (-1)^n/2^n)/9, n >= 0, by induction.
a(n) = a(n-1) + 2*a(n-2) + 2^(n-1), n >= 0, with input a(-2) = 1/4 and a(-1) = 0. See also A127984. (End)
E.g.f.: (exp(2*x)*(1 + 6*x) - cosh(x) + sinh(x))/9. - Stefano Spezia, Apr 09 2025
a(n) = Sum_{k=0..n+2} k * A238343(n+2,k). - Alois P. Heinz, Apr 30 2025

Simpler description from Vladeta Jovovic, Jul 18 2002

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User: Edward Early

A058886 Sum of the row of the character table of S_n corresponding to the partition 2,1^{n-2}.

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A076269 Size of largest antichain in partition lattice Par(n).

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

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A058884 Partial sums of the partition function (A000041), with the last term subtracted. Also the sum of the row of the character table for S_n corresponding to the partition n-1,1 for n>1. Also the sum over all partitions lambda of n of one less than the number of 1's in lambda.

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A045883 a(n) = ((3*n+1)*2^n - (-1)^n)/9.

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Magma

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A045883 a(n) = ((3n+1)2^n - (-1)^n)/9.