A210969 -id:A210969 - cp's OEIS frontend

A210971 Triangle read by rows in which row n lists the region number of the parts of the k-th partition of n, with partitions reverse lexicographically ordered.

1, 3, 2, 6, 5, 3, 11, 10, 8, 9, 5, 18, 17, 15, 16, 12, 13, 7, 29, 28, 26, 27, 23, 24, 18, 28, 20, 21, 11

Offset: 1

Omar E. Pol, Jun 30 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

			For n = 5 we have:
------------------------------------------------------
.              Two arrangements             Sum of
k           of the partitions of 5        partition k
------------------------------------------------------
7      [5]                          [5]        5
6      [3+2]                      [3+2]        5
5      [4+1]                    [4  +1]        5
4      [2+1+1]                [2+2  +1]        5
3      [3+1+1]              [3  +1  +1]        5
2      [2+1+1+1]          [2+1  +1  +1]        5
1      [1+1+1+1+1]      [1+1+1  +1  +1]        5
------------------------------------------------------
.              Two arrangements
.           of the region numbers           Sum of
k           of the partitions of 5          zone k
------------------------------------------------------
7      [7]                          [7]        7
6      [6,7]                      [6,7]       13
5      [5,7]                    [5,  7]       12
4      [4,5,7]                [4,5,  7]       16
3      [3,5,7]              [3,  5,  7]       15
2      [2,3,5,7]          [2,3,  5,  7]       17
1      [1,2,3,5,7]      [1,2,3,  5,  7]       18
------------------------------------------------------
So row 5 of triangle gives: 18, 17, 15, 16, 12, 13, 7.
.
Triangle begins:
1;
3,2;
6,5,3;
11,10,8,9,5;
18,17,15,16,12,13,7;
29,28,26,27,23,24,18,28,20,21,11;

Omar E. Pol, Illustration of the seven regions of 5

Column 1 is A026905. Right border = row lengths = A000041, n>=1. Row sums give A210972.

Cf. A135010, A138121, A182703, A194446, A206437, A210969.

A210972 Sum of all region numbers of all parts of all partitions of n.

Original entry on oeis.org

1, 5, 14, 43, 98, 255, 532, 1201, 2413, 4968, 9427, 18475

Offset: 1

Omar E. Pol, Jun 30 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

			For n = 5 we have:
---------------------------------------------------
.              Two arrangements
k           of the partitions of 5
---------------------------------------------------
7      [5]                          [5]
6      [3+2]                      [3+2]
5      [4+1]                    [4  +1]
4      [2+1+1]                [2+2  +1]
3      [3+1+1]              [3  +1  +1]
2      [2+1+1+1]          [2+1  +1  +1]
1      [1+1+1+1+1]      [1+1+1  +1  +1]
---------------------------------------------------
.              Two arrangements
.           of the region numbers           Sum of
k           of the partitions of 5          zone k
---------------------------------------------------
7      [7]                          [7]        7
6      [6,7]                      [6,7]       13
5      [5,7]                    [5,  7]       12
4      [4,5,7]                [4,5,  7]       16
3      [3,5,7]              [3,  5,  7]       15
2      [2,3,5,7]          [2,3,  5,  7]       17
1      [1,2,3,5,7]      [1,2,3,  5,  7]       18
---------------------------------------------------
The total sum is a(5) = 1+2^2+3^2+4+5^2+6+7^2 = 1+4+9+4+25+6+49 = 18+17+15+16+12+13+7 = 98.

Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A210969. Row sums of triangle A210971.

Cf. A135010, A138121, A182703, A194446, A210437, A210966.

A210966 Sum of all region numbers of all parts of the n-th region of the shell model of partitions.

Original entry on oeis.org

1, 4, 9, 4, 25, 6, 49, 8, 18, 10, 121, 12, 26, 14, 225, 16, 34, 18, 76, 20, 21, 484, 23, 48, 25, 104, 27, 56, 29, 900, 31, 64, 33, 136, 35, 36, 259, 38, 78, 40, 41, 1764, 43, 88, 45, 184, 47, 96, 49, 400, 51, 52, 159, 54, 55, 3136, 57, 116, 59, 240

Offset: 1

Omar E. Pol, Jul 01 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

			The first seven regions of the shell model of partitions (or the seven regions of 5) are [1], [2, 1], [3, 1, 1], [2], [4, 2, 1, 1, 1], [3], [5, 2, 1, 1, 1, 1, 1] therefore the "region numbers" are [1], [2, 2], [3, 3, 3], [4], [5, 5, 5, 5, 5], [6], [7, 7, 7, 7, 7, 7, 7]. So a(1)..a(7) give: 1, 4, 9, 4, 25, 6, 49.
Also written as an irregular triangle the sequence begins:
1;
4;
9;
4,25;
6,49;
8,18,10,121;
12,26,14,225;
16,34,18,76,20,21,484;
23,48,25,104,27,56,29,900;
31,64,33,136,35,36,259,38,78,40,41,1764;
43,88,45,184,47,96,49,400,51,52,159,54,55,3136;

Omar E. Pol, Illustration of the seven regions of 5

Row n has length A187219(n). Row sums give A210969. Right border gives A001255, n >= 1.

Cf. A135010, A138121, A194446, A182703, A206437, A210971, A210972.

a(n) = n*A194446(n).

cp's OEIS Frontend

A210971 Triangle read by rows in which row n lists the region number of the parts of the k-th partition of n, with partitions reverse lexicographically ordered.

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A210972 Sum of all region numbers of all parts of all partitions of n.

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A210966 Sum of all region numbers of all parts of the n-th region of the shell model of partitions.

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