Peter J. Taylor - cp's OEIS frontend

A381575 Number of disjoint-union partial algebras with zero on [n].

1, 2, 7, 68, 4619, 15621334

Offset: 0

Peter J. Taylor, Feb 28 2025

A disjoint-union partial algebra on a set S is a subset of the power set of S which is closed under union of disjoint sets.
A disjoint-union partial algebra with zero on a set S is a disjoint-union partial algebra on S which contains the empty set.
There are twice as many disjoint-union partial algebras on S as disjoint-union partial algebras with zero on S because the disjoint-union partial algebras without the empty set can be placed in bijection with those which have the empty set.

Hirsch, R., & McLean, B. (2017). Disjoint-union partial algebras. Logical Methods in Computer Science, 13.

Robin Hirsch and Brett McLean, Disjoint-union partial algebras, arXiv:1612.00252 [math.RA], 2016-2017.

Cf. A380571, A381472 (unlabeled case).

def A381575(n):
    cnt=0
    for p in range(1,2**(2**n),2):
        for a in range(1,2**n):
            if p&(1<Bert Dobbelaere, Mar 16 2025

A380571 Number of Dynkin systems on [n].

Original entry on oeis.org

1, 1, 2, 5, 19, 137, 3708, 1506404, 230328505024

Offset: 0

Peter J. Taylor, Feb 24 2025

A Dynkin system on a set S is a subset of the power set of S which contains the empty set, is closed under complements in S, and is closed under union of disjoint sets.

			The a(3) = 5 systems are:
  {{}, {1,2,3}}
  {{}, {1}, {2,3}, {1,2,3}}
  {{}, {2}, {1,3}, {1,2,3}}
  {{}, {3}, {1,2}, {1,2,3}}
  {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
The a(4) = 19 systems are 15 sigma-algebras counted by A000110(4) and 4 other systems:
  {{}, {1,2,3,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}
  {{}, {1,2,3,4}, {1,2}, {1,3}, {2,4}, {3,4}}
  {{}, {1,2,3,4}, {1,2}, {1,4}, {2,3}, {3,4}}
  {{}, {1,2,3,4}, {1,3}, {1,4}, {2,3}, {2,4}}

Martin Rubey and Peter Taylor, What is the number of finite Dynkin systems?, MathOverflow.
Wikipedia, Dynkin system

Cf. A000110, A102894, A381471 (unlabeled case).

a(n) >= A000110(n).

A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n.

Original entry on oeis.org

0, 1, 2, 6, 18, 30, 576, 840, 24480, 93696, 800640, 7983360, 65318400, 2286926400, 13680979200, 797369149440, 16753029012720, 10176199188480, 159943859712000, 26453863460044800, 470500040794291200, 20720967220237197312, 61690805562507264000

Offset: 0

Peter J. Taylor, Apr 04 2022

nonn

a(n) >= A038667(2*n).
Conjecture: a(n) = A038667(2*n) for all n. It is verified for n<=70. - Max Alekseyev, Jun 18 2022
Bernardo Recamán Santos proposes that this should be called Luciana's sequence for the student whose question prompted its investigation. (See MathOverflow link below.)

			For n = 4, the partition A = {1,5,6,7} and B = {2,3,4,8} is optimal, giving difference 1*5*6*7 - 2*3*4*8 = 18.
_Rob Pratt_ computed the optimal solutions for n <= 10:
[ n]    a(n)                   partitions of 2n
------------------------------------------------------------------
[ 1]       1                         2 | 1
[ 2]       2                       2,3 | 1,4
[ 3]       6                     1,5,6 | 2,3,4
[ 4]      18                   1,5,6,7 | 2,3,4,8
[ 5]      30                2,3,4,8,10 | 1,5,6,7,9
[ 6]     576              1,4,7,8,9,11 | 2,3,5,6,10,12
[ 7]     840           2,4,5,6,8,11,14 | 1,3,7,9,10,12,13
[ 8]   24480        1,5,6,7,8,13,14,15 | 2,3,4,9,10,11,12,16
[ 9]   93696     2,3,6,8,9,11,12,13,18 | 1,4,5,7,10,14,15,16,17
[10]  800640  2,3,4,8,9,11,12,18,19,20 | 1,5,6,7,10,13,14,15,16,17

Max Alekseyev, Table of n, a(n) for n = 0..70
Gordon Hamilton, Thirsty Fractions, MathPickle, 2013, for elementary and middle school teachers.
MathOverflow discussion Splitting the integers from 1 to 2n into two sets with products as close as possible.

Cf. A061057, A038667.

from math import prod, factorial
from itertools import combinations
def A352813(n):
    m = factorial(2*n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for d in combinations(range(2,2*n+1),n-1)) # Chai Wah Wu, Apr 06 2022

def A352813(n):
    return min(abs(prod(A)-prod(B)) for (A,B) in SetPartitions((1..2*n), [n,n]))
[A352813(n) for n in (1..10)] # Freddy Barrera, Apr 05 2022

A332755 Lapidary numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 31, 45, 61, 87, 119, 171, 233, 334, 459, 655, 904, 1288, 1782, 2535, 3517, 4995, 6935, 9848, 13703, 19437, 27070, 38376, 53528, 75842, 105878, 149966, 209555, 296707, 414922, 587304, 821853, 1163052, 1628574, 2304082

Offset: 0

Peter J. Taylor, Feb 22 2020

nonn

Consider a two-player stone-throwing game with a single shared pile of stones. The players alternately remove one or more stones from the pile until it is empty. In addition, each player seeks to communicate a message through their sequence of moves. If there are initially n stones then a(n) is the largest number m such that both players can communicate at least m distinct messages.

For n > 0, a(n) is also the size of the Durfee square of the partition defined in A064660.

			For n=4, one strategy which allows both players to communicate one of two messages is each remove one or two stones on their first turn.

Peter J. Taylor, Table of n, a(n) for n = 0..60
Peter J. Taylor, The lapidary numbers, or the combinatorics of communication by throwing stones (preprint).
Peter J. Taylor, The lapidary numbers, or the combinatorics of communication by throwing stones, Eureka, 65 (2018), pp. 89-90.
Peter J. Taylor, Python program

Cf. A064660.

Asymptotically, a(n) is within a subexponential factor of 2^(n/2).

A259049 Number of self-complementary plane partitions in a (2n)-cube.

Original entry on oeis.org

1, 4, 400, 960400, 54218191104, 71410553858811024, 2186315392560559723530496, 1552832545847343203950118294425600, 25554649541466337940020968722797075170918400, 9736551559782513812975251884508283964266367033264640000

Offset: 0

Peter J. Taylor, Jun 17 2015

Odd cubes have no self-complementary plane partitions.

R. P. Stanley, Symmetries of Plane Partitions, J. Comb. Theory Ser. A 43 (1986), 103-113.
P. J. Taylor, Counting distinct dimer hex tilings, Preprint, 2015.

Cf. A008793.

a(n) = prod(i=0, n-1, i!^2*(i+2*n)!^2 / (i+n)!^4) \\ Michel Marcus, Jun 18 2015

a(n) = Product_{i=0..n-1} i!^2 (i+2n)!^2 / (i+n)!^4.

a(n) = A008793(n)^2.

A065109 Triangle T(n,k) of coefficients relating to Bezier curve continuity.

Original entry on oeis.org

1, 2, -1, 4, -4, 1, 8, -12, 6, -1, 16, -32, 24, -8, 1, 32, -80, 80, -40, 10, -1, 64, -192, 240, -160, 60, -12, 1, 128, -448, 672, -560, 280, -84, 14, -1, 256, -1024, 1792, -1792, 1120, -448, 112, -16, 1, 512, -2304, 4608, -5376, 4032, -2016, 672, -144, 18, -1, 1024, -5120, 11520, -15360, 13440

Offset: 0

Peter J. Taylor, Nov 12 2001

Row sums are 1, antidiagonal sums are the natural numbers. - Gerald McGarvey, May 29 2005
Row sums = 1. - Roger L. Bagula, Sep 12 2008
Riordan array (1/(1-2x), -x/(1-2x)). - Philippe Deléham, Nov 27 2009
Triangle T(n,k), read by rows, given by [2,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2009

			For C-2 continuity between P and Q we require Q_0 = P_n; Q_1 = 2P_n - P_n-1; Q_2 = 4P_n - 4P_n-1 + P_n-2.
Triangle begins:
     1;
     2,     -1;
     4,     -4,     1;
     8,    -12,     6,     -1;
    16,    -32,    24,     -8,     1;
    32,    -80,    80,    -40,    10,     -1;
    64,   -192,   240,   -160,    60,    -12,     1;
   128,   -448,   672,   -560,   280,    -84,    14,    -1;
   256,  -1024,  1792,  -1792,  1120,   -448,   112,   -16,    1;
   512,  -2304,  4608,  -5376,  4032,  -2016,   672,  -144,   18,   -1;
  1024,  -5120, 11520, -15360, 13440,  -8064,  3360,  -960,  180,  -20,  1;
  2048, -11264, 28160, -42240, 42240, -29568, 14784, -5280, 1320, -220, 22, -1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Peter J. Taylor, Conditions for C-a Continuity of Bezier Curves

Cf. A038207, A013609. Apart from signs, same as A038207.

a065109 n k = a065109_tabl !! n !! k
a065109_row n = a065109_tabl !! n
a065109_tabl = iterate
   (\row -> zipWith (-) (map (* 2) row ++ [0]) ([0] ++ row)) [1]
-- Reinhard Zumkeller, Apr 25 2013

/* As triangle: */  [[(-1)^k*2^(n-k)*Binomial(n, k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Apr 26 2015

seq(seq((-1)^k * 2^(n-k) * binomial(n, k), k= 0 .. n), n = 0 .. 12); # Robert Israel, Apr 26 2015

t[n_, m_, k_] = (-1)^m*Multinomial[n - m - k, m, k]; Table[Table[Sum[t[n, m, k], {k, 0, n}], {m, 0, n}], {n, 0, 11}]; Flatten[%] (* Roger L. Bagula, Sep 12 2008 *)
Flatten[Table[(-1)^k 2^(n-k) Binomial[n,k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 13 2013 *)

T(n, k) = (-1)^k * 2^(n-k) * binomial(n, k).
Sum_{i=0..n} binomial(n,i) * (-1)^i * T(i,r) = (-1)^(n-r) * binomial(n,r).
For n > 0, T(n, k) = 2*T(n-1, k) - T(n-1, k-1). - Gerald McGarvey, May 29 2005
p(n,m,k) = (-1)^m*multinomial(n - m - k, m, k); t(n,m) = Sum_{k=0..n} (-1)^m*multinomial(n - m - k, m, k). - Roger L. Bagula, Sep 12 2008
Sum_{k=0..n} T(n,k)*A000108(k) = A001405(n). - Philippe Deléham, Nov 27 2009
Sum_{k=0..n} T(n,k)*x^k = (2-x)^n. - Philippe Deléham, Dec 15 2009
G.f.: Sum_{n>=0} (2-x)^n * x^(n*(n+1)/2). - Robert Israel, Apr 26 2015
G.f.: 1/(1-2*x+x*y). - R. J. Mathar, Aug 11 2015

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User: Peter J. Taylor

A381575 Number of disjoint-union partial algebras with zero on [n].

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A380571 Number of Dynkin systems on [n].

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A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n.

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A332755 Lapidary numbers.

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A259049 Number of self-complementary plane partitions in a (2n)-cube.

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A065109 Triangle T(n,k) of coefficients relating to Bezier curve continuity.

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