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

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

1,6

Comments

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).

Links

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

Example

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

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A325668 A070202 A280129 * A130207 A325433 A167688

Adjacent sequences: A227341 A227342 A227343 * A227345 A227346 A227347

Keyword

nonn,tabl

Author

Joerg Arndt, Jul 08 2013

Status

approved