A227345 -id:A227345 - cp's OEIS frontend

A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 5, 2, 0, 1, 5, 4, 0, 1, 5, 6, 0, 1, 6, 7, 1, 0, 1, 6, 10, 1, 0, 1, 7, 11, 3, 0, 1, 9, 13, 4, 0, 1, 7, 18, 6, 0, 1, 8, 20, 9, 0, 1, 10, 21, 14, 0, 1, 9, 27, 16, 1, 0, 1, 10, 29, 22, 2

Offset: 0

Alois P. Heinz, Jul 16 2013

The boundary size is the number of parts having fewer than two neighbors.

			T(12,1) = 1: [12].
T(12,2) = 6: [1,11], [2,10], [3,4,5], [3,9], [4,8], [5,7].
T(12,3) = 7: [1,2,3,6], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,3,7], [2,4,6].
T(12,4) = 1: [1,2,4,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3, 1;
  0, 1, 3, 2;
  0, 1, 5, 2;
  0, 1, 5, 4;
  0, 1, 5, 6;
  0, 1, 6, 7, 1;

Alois P. Heinz, Rows n = 0..600, flattened

Columns k=0-10 give: A000007, A057427, A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567.
Row sums give: A000009.
Last elements of rows give: A227552.
Cf. A227345 (a version with trailing zeros), A053993, A201077, A227568, A224878 (one part of size 0 allowed).

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=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..30);

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, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 12 2016, after Alois P. Heinz *)

A227344 Triangle read by rows, partitions into distinct parts by perimeter.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 0, 0, 7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 16, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 20, 0

Offset: 1

Joerg Arndt, Jul 08 2013

The perimeter of a partition is the sum of all parts p that do not have two neighbors (that is, not both p-1 and p+1 are parts).
Row sums are A000009.
Column sums are A122129 (noted by Patrick Devlin).

			Triangle starts (dots for zeros):
01: 1
02: . 1
03: . . 2
04: . . . 2
05: . . . . 3
06: . . . 1 . 3
07: . . . . . . 5
08: . . . . . . . 6
09: . . . . . 1 . . 7
10: . . . . 1 . . . . 9
11: . . . . . . . . 1 . 11
12: . . . . . . . 1 . 1 . 13
13: . . . . . . . . 1 . 1 . 16
14: . . . . . . 1 . . . . 1 . 20
15: . . . . . 1 . . . 1 . 1 1 . 23
16: . . . . . . . . . . 2 . 1 1 . 28
17: . . . . . . . . . . . 2 . 1 2 . 33
18: . . . . . . . . 1 . . 1 2 . 1 2 . 39
19: . . . . . . . . . 1 . . 1 1 1 1 3 . 46
20: . . . . . . . 1 . . . . . 1 1 2 1 3 . 55
21: . . . . . . 1 . . . . . . 2 2 1 2 1 4 . 63
22: . . . . . . . . . . 1 . 1 . 2 1 1 2 2 4 . 75
23: . . . . . . . . . . . 1 . 1 . 2 1 3 2 2 5 . 87
24: . . . . . . . . . . . . 1 . 1 2 3 . 4 2 3 5 . 101
25: . . . . . . . . . 1 . . . 1 . 1 1 3 . 6 2 3 7 . 117
26: . . . . . . . . . . 1 . 1 . . . 2 1 3 . 7 2 4 8 . 136
27: . . . . . . . . 1 . . . . 1 . . . 5 2 2 1 8 3 4 9 . 156
28: . . . . . . . 1 . . . . . . 1 1 . . 4 2 3 2 8 4 5 11 . 180
29: . . . . . . . . . . . . . . 1 2 1 . . 4 3 3 3 9 5 5 13 . 207
30: . . . . . . . . . . . 1 . . 1 1 1 1 . 3 6 2 2 5 9 6 6 14 . 238

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

Cf. A227345 (partitions by boundary size).
Cf. A227426 (diagonal: number of partitions with maximal perimeter).
Cf. A227538 (smallest k with positive T(n,k)), A227614 (second lower diagonal). - Alois P. Heinz, Jul 17 2013

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x^(i+1), 1),
      expand(`if`(i<1, 0, `if`(t>1, x^(i+1), 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x^(i+1), 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^(i+1), 1], Expand[If[i<1, 0, If[t>1, x^(i+1), 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t == 2, x^(i+1), 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, Jan 28 2015, after Alois P. Heinz *)

A227552 Number of partitions of n into distinct parts with maximal boundary size.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 4, 6, 1, 1, 3, 4, 6, 9, 14, 1, 2, 3, 5, 8, 11, 17, 24, 1, 1, 3, 5, 8, 11, 18, 24, 35, 49, 1, 2, 3, 6, 9, 14, 21, 30, 42, 60, 81, 1, 1, 3, 5, 9, 13, 21, 29, 43, 60, 84, 113, 156, 1, 2, 3, 6, 10, 15, 24, 35, 50, 71, 99, 134, 184, 246

Offset: 0

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having less than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Last elements of rows of A227551.
Last nonzero elements of rows of A227345.
Cf. A053993, A201077.

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:
a:= n-> (p->coeff(p, x, degree(p)))(b(n$2, 0)):
seq(a(n), n=0..100);

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]]]]]; a[n_] := Function [p, Coefficient[p, x, Exponent[p, x]]][b[n, n, 0]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = A227551(n,A227568(n)).

A227559 Number of partitions of n into distinct parts with boundary size 2.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 5, 5, 5, 6, 6, 7, 9, 7, 8, 10, 9, 10, 12, 11, 11, 12, 13, 13, 15, 14, 14, 17, 15, 15, 18, 17, 19, 19, 18, 19, 21, 20, 20, 23, 21, 22, 26, 23, 23, 24, 25, 26, 27, 26, 26, 29, 29, 28, 30, 29, 29, 32, 30, 31, 35, 31, 34, 35, 33, 34, 36, 37, 35

Offset: 3

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=2 of A227345, A227551.

A227560 Number of partitions of n into distinct parts with boundary size 3.

Original entry on oeis.org

1, 2, 2, 4, 6, 7, 10, 11, 13, 18, 20, 21, 27, 29, 32, 37, 42, 45, 49, 54, 58, 65, 70, 72, 82, 87, 90, 98, 103, 111, 119, 124, 130, 139, 147, 151, 163, 169, 174, 187, 196, 203, 211, 219, 229, 240, 250, 256, 268, 279, 288, 300, 311, 318, 333, 342, 350, 368, 376

Offset: 7

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=3 of A227345, A227551.

A227561 Number of partitions of n into distinct parts with boundary size 4.

Original entry on oeis.org

1, 1, 3, 4, 6, 9, 14, 16, 22, 28, 35, 42, 53, 62, 73, 86, 98, 114, 134, 148, 168, 192, 212, 235, 264, 289, 320, 355, 385, 419, 461, 495, 538, 586, 626, 674, 734, 779, 835, 898, 951, 1013, 1087, 1147, 1219, 1300, 1367, 1447, 1542, 1614, 1701, 1801, 1887, 1982

Offset: 12

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 12..1000

Column k=4 of A227345, A227551.

A227562 Number of partitions of n into distinct parts with boundary size 5.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 17, 24, 31, 43, 54, 67, 86, 108, 129, 160, 192, 224, 268, 315, 360, 424, 487, 556, 637, 723, 810, 921, 1029, 1141, 1285, 1426, 1568, 1746, 1920, 2102, 2314, 2529, 2748, 3013, 3269, 3533, 3848, 4164, 4481, 4840, 5214, 5590, 6016, 6448, 6882

Offset: 19

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 19..1000

Column k=5 of A227345, A227551.

A227563 Number of partitions of n into distinct parts with boundary size 6.

Original entry on oeis.org

1, 1, 3, 5, 8, 11, 18, 24, 35, 49, 64, 83, 112, 138, 177, 220, 272, 333, 409, 486, 586, 699, 817, 962, 1131, 1306, 1515, 1748, 1999, 2286, 2610, 2946, 3337, 3770, 4219, 4729, 5297, 5898, 6553, 7279, 8042, 8882, 9803, 10755, 11817, 12966, 14152, 15454, 16877

Offset: 27

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 27..1000

Column k=6 of A227345, A227551.

A227564 Number of partitions of n into distinct parts with boundary size 7.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 21, 30, 42, 60, 81, 107, 145, 186, 238, 303, 382, 474, 591, 723, 880, 1068, 1285, 1528, 1829, 2158, 2534, 2970, 3472, 4022, 4668, 5377, 6173, 7076, 8076, 9168, 10428, 11793, 13288, 14971, 16814, 18807, 21050, 23467, 26099, 29014, 32158

Offset: 37

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 37..1000

Column k=7 of A227345, A227551.

A227565 Number of partitions of n into distinct parts with boundary size 8.

Original entry on oeis.org

1, 1, 3, 5, 9, 13, 21, 29, 43, 60, 84, 113, 156, 201, 266, 346, 443, 558, 709, 877, 1091, 1343, 1643, 1994, 2423, 2898, 3470, 4139, 4897, 5773, 6800, 7940, 9266, 10779, 12466, 14393, 16590, 19009, 21756, 24847, 28253, 32073, 36354, 41035, 46275, 52088, 58430

Offset: 48

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 48..1000

Column k=8 of A227345, A227551.

cp's OEIS Frontend

A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

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A227344 Triangle read by rows, partitions into distinct parts by perimeter.

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A227552 Number of partitions of n into distinct parts with maximal boundary size.

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A227559 Number of partitions of n into distinct parts with boundary size 2.

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A227560 Number of partitions of n into distinct parts with boundary size 3.

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A227561 Number of partitions of n into distinct parts with boundary size 4.

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A227562 Number of partitions of n into distinct parts with boundary size 5.

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A227563 Number of partitions of n into distinct parts with boundary size 6.

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A227564 Number of partitions of n into distinct parts with boundary size 7.

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A227565 Number of partitions of n into distinct parts with boundary size 8.

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