A227344 -id:A227344 - cp's OEIS frontend

A227345 Triangle read by rows, partitions into distinct parts by size of boundary.

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 5, 2, 0, 0, 0, 0, 0, 0, 1, 5, 4, 0, 0, 0, 0, 0, 0, 0, 1, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 11, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 13, 4, 0

Offset: 1

Joerg Arndt, Jul 08 2013

The boundary size of a partition is the number of parts p that do not have two neighbors (that is, not both p-1 and p+1 are parts).
Row sums are A000009.
Conjecture: there exists a partition (into distinct parts) of n with boundary size k if and only if 0 < k^2 * 3/4 <= n. - Patrick Devlin, Jul 13 2013

			Triangle starts (dots for zeros, trailing zeros omitted for n>=14):
  01: 1
  02: 1 .
  03: 1 1 .
  04: 1 1 . .
  05: 1 2 . . .
  06: 1 3 . . . .
  07: 1 3 1 . . . .
  08: 1 3 2 . . . . .
  09: 1 5 2 . . . . . .
  10: 1 5 4 . . . . . . .
  11: 1 5 6 . . . . . . . .
  12: 1 6 7 1 . . . . . . . .
  13: 1 6 10 1 . . . . . . . . .
  14: 1 7 11 3 . . . . . . . . .
  15: 1 9 13 4 . . . . . . . . .
  16: 1 7 18 6 . . . . . . . . .
  17: 1 8 20 9 . . . . . . . . .
  18: 1 10 21 14 . . . . . . . .
  19: 1 9 27 16 1 . . . . . . .
  20: 1 10 29 22 2 . . . . . . .
  21: 1 12 32 28 3 . . . . . . .
  22: 1 11 37 35 5 . . . . . . .
  23: 1 11 42 42 8 . . . . . . .
  24: 1 12 45 53 11 . . . . . .
  25: 1 13 49 62 17 . . . . . .
  26: 1 13 54 73 24 . . . . . .
  27: 1 15 58 86 31 1 . . . . .
  28: 1 14 65 98 43 1 . . . . .
  29: 1 14 70 114 54 3 . . . . .
  30: 1 17 72 134 67 5 . . . . .
  31: 1 15 82 148 86 8 . . . . .
  32: 1 15 87 168 108 11 . . . .
  33: 1 18 90 192 129 18 . . . .
  34: 1 17 98 212 160 24 . . . .
  35: 1 19 103 235 192 35 . . .
  36: 1 19 111 264 224 49 . . .
  37: 1 18 119 289 268 64 1 . .
  38: 1 19 124 320 315 83 2 . .
  39: 1 21 130 355 360 112 3 . .
  40: 1 20 139 385 424 138 6 . .
In particular, for the tenth row of this table, note that the partitions of ten into distinct parts are 10 = 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 4 + 3 + 2 + 1 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2. These partitions are sorted by increasing number of parts in the boundary. In particular, note that 4 + 3 + 2 + 1 has only two parts in its boundary (namely 4 and 1). - _Patrick Devlin_, Jul 13 2013

Joerg Arndt, Table of n, a(n) for n = 1..5050

Cf. A227344 (partitions by perimeter).

Columns k=1-10 give: A057427 (for n>=1), A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567. Cf. A227551 (a version without trailing zeros), A227552. - Alois P. Heinz, Jul 16 2013

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
      expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
seq(T(n), n=1..20);  # Alois P. Heinz, Jul 16 2013

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t>1, x, 1], Expand[If[i<1, 0, If[t>1, x, 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t == 2, x, 1] * b[n-i, i-1, Quotient[t, 2]+2]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n, 0]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

From Patrick Devlin, Jul 13 2013: (Start)
Let a(n,k) denote the number of partitions into distinct parts of n with boundary size k. Then for all n>0 and k>=0, we have a(n,k+1) >= floor(binomial(n-k, k) * 2^(-binomial(k, 2))) = floor(binomial(n-k, k) * 2^(-A000217(k))). (Proof is by noting a(n,k) >= Sum_{j=1..(n/2-1)} a(j,k-1).)
On the other hand, for all n>0 and k>=0, we also have that a(n,k+1) <= binomial(n-k,k)*A000045(k+1). This is obtained by considering the largest k parts of the boundary, which must be some subset of {1, 2, ..., n-k}. Then the possible 'gaps' of the boundary can each either be filled with the corresponding consecutive integers or left empty. (End)

A227538 Smallest k such that a partition of n into distinct parts with perimeter k exists.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 7, 8, 6, 5, 9, 8, 9, 7, 6, 11, 12, 9, 10, 8, 7, 11, 12, 13, 10, 11, 9, 8, 15, 12, 13, 14, 11, 12, 10, 9, 16, 17, 13, 14, 15, 12, 13, 11, 10, 16, 17, 18, 14, 15, 16, 13, 14, 12, 11, 16, 17, 18, 19, 15, 16, 17, 14, 15, 13, 12, 22, 17, 18, 19

Offset: 0

Alois P. Heinz, Jul 16 2013

The perimeter is the sum of all parts having less than two neighbors.

a(n) is also the smallest perimeter among all sets of positive integers whose volume (sum) is n. - Patrick Devlin, Jul 23 2013

			a(0) = 0: the empty partition [] has perimeter 0.
a(1) = 1: [1] has perimeter 1.
a(3) = 3: [1,2], [3] have perimeter 3.
a(6) = 4: [1,2,3] has perimeter 4.
a(7) = 7: [1,2,4], [3,4], [2,5], [1,6], [7] have perimeter 7; no partition of 7 into distinct parts has a smaller perimeter.
a(10) = 5: [1,2,3,4] has perimeter 5.
a(15) = 6: [1,2,3,4,5] has perimeter 6.
a(29) = 15: [1,2,3,4,5,6,8] has perimeter 1+6+8 = 15.
a(30) = 12: [4,5,6,7,8] has perimeter 12.

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

Cf. A227344, A186053 (smallest perimeter among all sets of nonnegative integers).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, i+1, 0),
      `if`(i<1, infinity, min(`if`(t>1, i+1, 0)+b(n, i-1, iquo(t, 2)),
      `if`(i>n, NULL, `if`(t=2, i+1, 0)+b(n-i, i-1, iquo(t, 2)+2)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t>1, i+1, 0], If[i<1, Infinity, Min[If[t>1, i+1, 0] + b[n, i-1, Quotient[t, 2]], If[i>n, Infinity, If[t == 2, i+1, 0] + b[n-i, i-1, Quotient[t, 2]+2]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = min { k : A227344(n,k) > 0 }.

a(A000217(n)) = n+1 for n>1.

A227426 Number of partitions into distinct parts without three consecutive parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 7, 9, 11, 13, 16, 20, 23, 28, 33, 39, 46, 55, 63, 75, 87, 101, 117, 136, 156, 180, 207, 238, 272, 311, 355, 404, 460, 522, 592, 670, 758, 855, 965, 1087, 1223, 1373, 1543, 1728, 1936, 2166, 2421, 2702, 3016, 3359, 3741, 4162, 4626, 5136, 5702, 6320, 7002, 7753, 8576, 9479, 10473

Offset: 0

Joerg Arndt, Jul 11 2013

nonn

Number of partitions into distinct parts with maximal perimeter.

For n>=1, diagonal of A227344.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009.

a227426 = p 1 1 where
  p   0 = 1
  p k i m = if m < k then 0 else p (k + i) (3 - i) (m - k) + p (k + 1) 1 m
-- Reinhard Zumkeller, Jul 14 2013

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1, 0)+`if`(i>n or t=2, 0, b(n-i, i-1, t+1))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Jul 15 2013

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1, If[i<1, 0, b[n, i-1, 0] + If[i>n || t==2, 0, b[n-i, i-1, t+1]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

a(n) = c * exp(r*sqrt(n)) / n^(3/4), where r = 1.75931899568... and c = 0.2080626386... - Vaclav Kotesovec, May 24 2018

A227614 Number of partitions of n into distinct parts with perimeter n-2.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 8, 9, 11, 13, 14, 17, 19, 21, 25, 28, 32, 37, 42, 47, 55, 61, 69, 78, 88, 98, 112, 124, 140, 157, 176, 196, 221, 245, 274, 305, 340, 377, 420, 465, 517, 573, 634, 702, 777, 858, 949, 1047, 1154, 1273

Offset: 6

Alois P. Heinz, Jul 17 2013

nonn

The perimeter is the sum of all parts having less than two neighbors.

a(n) counts all partitions of n into distinct parts where only part 2 has two neighbors.

			a(6) = 1: [1,2,3].
a(11) = 1: [1,2,3,5].
a(17) = 2: [1,2,3,5,6], [1,2,3,11].
a(19) = 3: [1,2,3,5,8], [1,2,3,6,7], [1,2,3,13].
a(21) = 4: [1,2,3,7,8], [1,2,3,5,10], [1,2,3,6,9], [1,2,3,15].
a(23) = 5: [1,2,3,5,12], [1,2,3,6,11], [1,2,3,7,10], [1,2,3,8,9], [1,2,3,17].

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Cf. A227344.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<5, 0,
       b(n, i-1, 0)+`if`(i>n or t=2, 0, b(n-i, i-1, t+1))))
    end:
a:= n-> b(n-6, n-6, 0):
seq(a(n), n=6..100);

b[n_, i_, t_] := b[n, i, t] = If[n==0, 1, If[i<5, 0, b[n, i-1, 0] + If[i>n || t==2, 0, b[n-i, i-1, t+1]]]]; a[n_] := b[n-6, n-6, 0]; Table[a[n], {n, 6, 100}] (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

a(n) = A227344(n,n-2).

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A227345 Triangle read by rows, partitions into distinct parts by size of boundary.

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A227538 Smallest k such that a partition of n into distinct parts with perimeter k exists.

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A227426 Number of partitions into distinct parts without three consecutive parts.

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A227614 Number of partitions of n into distinct parts with perimeter n-2.

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