Partition related number triangles

## Set partitions by number of blocks

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All partitions of a set with n elements into k blocks are enumerated by the number triangle called Stirling numbers of the second kind. When some partitions are left out or treated as equivalent the new set of partitions is enumerated by a new number triangle. The number triangles for noncrossing partitions are symmetric.

| All  | Noncrossing

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All  |

Triangle A008277 (Stirling2) with row sums A000110 (Bell)  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 3 | 1 |  |  |  |  |  |  |  |  |  |  | 5
4  | 1 | 7 | 6 | 1 |  |  |  |  |  |  |  |  |  | 15
5  | 1 | 15 | 25 | 10 | 1 |  |  |  |  |  |  |  |  | 52
6  | 1 | 31 | 90 | 65 | 15 | 1 |  |  |  |  |  |  |  | 203
7  | 1 | 63 | 301 | 350 | 140 | 21 | 1 |  |  |  |  |  |  | 877
8  | 1 | 127 | 966 | 1701 | 1050 | 266 | 28 | 1 |  |  |  |  |  | 4140
9  | 1 | 255 | 3025 | 7770 | 6951 | 2646 | 462 | 36 | 1 |  |  |  |  | 21147
10  | 1 | 511 | 9330 | 34105 | 42525 | 22827 | 5880 | 750 | 45 | 1 |  |  |  | 115975
11  | 1 | 1023 | 28501 | 145750 | 246730 | 179487 | 63987 | 11880 | 1155 | 55 | 1 |  |  | 678570
12  | 1 | 2047 | 86526 | 611501 | 1379400 | 1323652 | 627396 | 159027 | 22275 | 1705 | 66 | 1 |  | 4213597
13  | 1 | 4095 | 261625 | 2532530 | 7508501 | 9321312 | 5715424 | 1899612 | 359502 | 39325 | 2431 | 78 | 1 | 27644437

Diagonal: A129506

The 1,7,6,1 partitions of a 4-element set
with 1,2,3,4 blocks Triangle A001263 (Narayana) with row sums A000108 (Catalan)  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 3 | 1 |  |  |  |  |  |  |  |  |  |  | 5
4  | 1 | 6 | 6 | 1 |  |  |  |  |  |  |  |  |  | 14
5  | 1 | 10 | 20 | 10 | 1 |  |  |  |  |  |  |  |  | 42
6  | 1 | 15 | 50 | 50 | 15 | 1 |  |  |  |  |  |  |  | 132
7  | 1 | 21 | 105 | 175 | 105 | 21 | 1 |  |  |  |  |  |  | 429
8  | 1 | 28 | 196 | 490 | 490 | 196 | 28 | 1 |  |  |  |  |  | 1430
9  | 1 | 36 | 336 | 1176 | 1764 | 1176 | 336 | 36 | 1 |  |  |  |  | 4862
10  | 1 | 45 | 540 | 2520 | 5292 | 5292 | 2520 | 540 | 45 | 1 |  |  |  | 16796
11  | 1 | 55 | 825 | 4950 | 13860 | 19404 | 13860 | 4950 | 825 | 55 | 1 |  |  | 58786
12  | 1 | 66 | 1210 | 9075 | 32670 | 60984 | 60984 | 32670 | 9075 | 1210 | 66 | 1 |  | 208012
13  | 1 | 78 | 1716 | 15730 | 70785 | 169884 | 226512 | 169884 | 70785 | 15730 | 1716 | 78 | 1 | 742900

Diagonal: A000891

The 1,6,6,1 n. p. of a 4-element set
with 1,2,3,4 blocks

rot  |

Triangle A152175 with row sums A084423 |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 1 |  |  |  |  |  |  |  |  | 3
4  | 1 | 3 | 2 | 1 |  |  |  |  |  |  |  | 7
5  | 1 | 3 | 5 | 2 | 1 |  |  |  |  |  |  | 12
6  | 1 | 7 | 18 | 13 | 3 | 1 |  |  |  |  |  | 43
7  | 1 | 9 | 43 | 50 | 20 | 3 | 1 |  |  |  |  | 127
8  | 1 | 19 | 126 | 221 | 136 | 36 | 4 | 1 |  |  |  | 544
9  | 1 | 29 | 339 | 866 | 773 | 296 | 52 | 4 | 1 |  |  | 2361
10  | 1 | 55 | 946 | 3437 | 4281 | 2303 | 596 | 78 | 5 | 1 |  | 11703
11  | 1 | 93 | 2591 | 13250 | 22430 | 16317 | 5817 | 1080 | 105 | 5 | 1 | 61690
Triangle A209805 with row sums A054357 |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 1 |  |  |  |  |  |  |  |  |  |  | 3
4  | 1 | 2 | 2 | 1 |  |  |  |  |  |  |  |  |  | 6
5  | 1 | 2 | 4 | 2 | 1 |  |  |  |  |  |  |  |  | 10
6  | 1 | 3 | 10 | 10 | 3 | 1 |  |  |  |  |  |  |  | 28
7  | 1 | 3 | 15 | 25 | 15 | 3 | 1 |  |  |  |  |  |  | 63
8  | 1 | 4 | 26 | 64 | 64 | 26 | 4 | 1 |  |  |  |  |  | 190
9  | 1 | 4 | 38 | 132 | 196 | 132 | 38 | 4 | 1 |  |  |  |  | 546
10  | 1 | 5 | 56 | 256 | 536 | 536 | 256 | 56 | 5 | 1 |  |  |  | 1708
11  | 1 | 5 | 75 | 450 | 1260 | 1764 | 1260 | 450 | 75 | 5 | 1 |  |  | 5346
12  | 1 | 6 | 104 | 765 | 2736 | 5102 | 5102 | 2736 | 765 | 104 | 6 | 1 |  | 17428
13  | 1 | 6 | 132 | 1210 | 5445 | 13068 | 17424 | 13068 | 5445 | 1210 | 132 | 6 | 1 | 57148

Diagonal is probably A001246 (square Catalan numbers)
The calculation of T(13,6) is based on the assumption that T(13,7) = C(6)2. Matlab code from the calculation is here.

rot

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Triangle A152176 with row sums A084708 |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 1 |  |  |  |  |  |  |  |  | 3
4  | 1 | 3 | 2 | 1 |  |  |  |  |  |  |  | 7
5  | 1 | 3 | 5 | 2 | 1 |  |  |  |  |  |  | 12
6  | 1 | 7 | 14 | 11 | 3 | 1 |  |  |  |  |  | 37
7  | 1 | 8 | 31 | 33 | 16 | 3 | 1 |  |  |  |  | 93
8  | 1 | 17 | 82 | 137 | 85 | 27 | 4 | 1 |  |  |  | 354
9  | 1 | 22 | 202 | 478 | 434 | 171 | 37 | 4 | 1 |  |  | 1350
10  | 1 | 43 | 538 | 1851 | 2271 | 1249 | 338 | 54 | 5 | 1 |  | 6351
11  | 1 | 62 | 1401 | 6845 | 11530 | 8389 | 3056 | 590 | 70 | 5 | 1 | 31950

Triangle A209612 with row sums A111275 |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 1 |  |  |  |  |  |  |  |  | 3
4  | 1 | 2 | 2 | 1 |  |  |  |  |  |  |  | 6
5  | 1 | 2 | 4 | 2 | 1 |  |  |  |  |  |  | 10
6  | 1 | 3 | 8 | 8 | 3 | 1 |  |  |  |  |  | 24
7  | 1 | 3 | 12 | 17 | 12 | 3 | 1 |  |  |  |  | 49
8  | 1 | 4 | 19 | 41 | 41 | 19 | 4 | 1 |  |  |  | 130
9  | 1 | 4 | 27 | 78 | 116 | 78 | 27 | 4 | 1 |  |  | 336
10  | 1 | 5 | 38 | 148 | 298 | 298 | 148 | 38 | 5 | 1 |  | 980
11  | 1 | 5 | 50 | 250 | 680 | 932 | 680 | 250 | 50 | 5 | 1 | 2904

In the two illustrations above some set partitions, sharing the same colors, are grouped together.
They always correspond to the same integer partition, shown in the following illustration:

Triangle A008284 with row sums A000041 |  |  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |  | sum

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1  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 1 |  |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 1 |  |  |  |  |  |  |  |  |  |  | 3
4  | 1 | 2 | 1 | 1 |  |  |  |  |  |  |  |  |  | 5
5  | 1 | 2 | 2 | 1 | 1 |  |  |  |  |  |  |  |  | 7
6  | 1 | 3 | 3 | 2 | 1 | 1 |  |  |  |  |  |  |  | 11
7  | 1 | 3 | 4 | 3 | 2 | 1 | 1 |  |  |  |  |  |  | 15
8  | 1 | 4 | 5 | 5 | 3 | 2 | 1 | 1 |  |  |  |  |  | 22
9  | 1 | 4 | 7 | 6 | 5 | 3 | 2 | 1 | 1 |  |  |  |  | 30
10  | 1 | 5 | 8 | 9 | 7 | 5 | 3 | 2 | 1 | 1 |  |  |  | 42
11  | 1 | 5 | 10 | 11 | 10 | 7 | 5 | 3 | 2 | 1 | 1 |  |  | 56
12  | 1 | 6 | 12 | 15 | 13 | 11 | 7 | 5 | 3 | 2 | 1 | 1 |  | 77
13  | 1 | 6 | 14 | 18 | 18 | 14 | 11 | 7 | 5 | 3 | 2 | 1 | 1 | 101

Diagonal and second knight moves diagonal: A000041 from 0th entry
A008284 ⚪ A001844     =     A008284 ⚪ A081267     =     A000041

### Lah numbers

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Counting not only the partitions (which gives the Stirling2 numbers shown above), but also the possible permutations within the blocks gives the triangle of unsigned Lah numbers A105278 (with the row sums A000262).

Also here one could define a noncrossing version (where T(4;2) would be 32 instead of 36).

The unsigned Lah numbers

## Set partitions by type and by number of singletons

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| tabl triangles,
columns match number of singletons:  |  | A | N

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A | A124323 | A091867
Utr | A211356 | A211357
Utr&r | A211358 | A211359
tabf triangles,
columns match type:  |  | A | N  |     | Rc

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A  | A211350 | A211351 | A007318
Utr  | A211352 | A211353 | A047996
Utr&r  | A211354 | A211355 | A052307
Sequences of row sums:  |  | A | N

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A  | A000110 | A000108
Utr  | A084423 | A054357
Utr&r  | A084708 | A111275
Set partitions (up to rotation and reflection) by corresponding integer partition (their type) | Counting the displayed partitions in different ways leads to different number triangles.
These are examples for entry(6,3).

### All

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All, All        (by number of singletons)
Triangle A124323 with row sums A000110 (Bell)

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |  | sum

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0  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
1  | 0 | 1 |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 3 | 0 | 1 |  |  |  |  |  |  |  |  |  | 5
4  | 4 | 4 | 6 | 0 | 1 |  |  |  |  |  |  |  |  | 15
5  | 11 | 20 | 10 | 10 | 0 | 1 |  |  |  |  |  |  |  | 52
6  | 41 | 66 | 60 | 20 | 15 | 0 | 1 |  |  |  |  |  |  | 203
7  | 162 | 287 | 231 | 140 | 35 | 21 | 0 | 1 |  |  |  |  |  | 877
8  | 715 | 1296 | 1148 | 616 | 280 | 56 | 28 | 0 | 1 |  |  |  |  | 4140
9  | 3425 | 6435 | 5832 | 3444 | 1386 | 504 | 84 | 36 | 0 | 1 |  |  |  | 21147
10  | 17722 | 34250 | 32175 | 19440 | 8610 | 2772 | 840 | 120 | 45 | 0 | 1 |  |  | 115975
11  | 98253 | 194942 | 188375 | 117975 | 53460 | 18942 | 5082 | 1320 | 165 | 55 | 0 | 1 |  | 678570
12  | 580317 | 1179036 | 1169652 | 753500 | 353925 | 128304 | 37884 | 8712 | 1980 | 220 | 66 | 0 | 1 | 4213597

All, Noncrossing        (by number of singletons)
Triangle A091867 with row sums A000108 (Catalan)

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |  | sum

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0  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
1  | 0 | 1 |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 3 | 0 | 1 |  |  |  |  |  |  |  |  |  | 5
4  | 3 | 4 | 6 | 0 | 1 |  |  |  |  |  |  |  |  | 14
5  | 6 | 15 | 10 | 10 | 0 | 1 |  |  |  |  |  |  |  | 42
6  | 15 | 36 | 45 | 20 | 15 | 0 | 1 |  |  |  |  |  |  | 132
7  | 36 | 105 | 126 | 105 | 35 | 21 | 0 | 1 |  |  |  |  |  | 429
8  | 91 | 288 | 420 | 336 | 210 | 56 | 28 | 0 | 1 |  |  |  |  | 1430
9  | 232 | 819 | 1296 | 1260 | 756 | 378 | 84 | 36 | 0 | 1 |  |  |  | 4862
10  | 603 | 2320 | 4095 | 4320 | 3150 | 1512 | 630 | 120 | 45 | 0 | 1 |  |  | 16796
11  | 1585 | 6633 | 12760 | 15015 | 11880 | 6930 | 2772 | 990 | 165 | 55 | 0 | 1 |  | 58786
12  | 4213 | 19020 | 39798 | 51040 | 45045 | 28512 | 13860 | 4752 | 1485 | 220 | 66 | 0 | 1 | 208012

By type

All, All        (by type)        Triangle A211350

0   1   2    3   4    5   6     7     8    9  10    11    12    13  14    15     16     17    18    19    20  21     22    23     24   25    26    27    28  29    30     31     32    33     34     35   36     37   38    39   40  41     42    43     44    45     46    47    48  49    50    51   52   53   54  55    56    57     58    59    60    61   62     63   64    65   66  67    68   69    70  71   72  73  74  75  76

1,  1,
1,  3,  1,
1,  6,  4,   3,  1,
1, 10, 10,  15,  5,  10,  1,
1, 15, 20,  45, 15,  60,  6,   15,   15,  10,  1,
1, 21, 35, 105, 35, 210, 21,  105,  105,  70,  7,  105,   21,   35,  1,
1, 28, 56, 210, 70, 560, 56,  420,  420, 280, 28,  840,  168,  280,  8,  105,   210,   280,   28,   56,   35,  1,
1, 36, 84, 378,126,1260,126, 1260, 1260, 840, 84, 3780,  756, 1260, 36,  945,  1890,  2520,  252,  504,  315,  9,  1260,  378,  1260,  36,  280,   84,  126,  1,
1, 45,120, 630,210,2520,252, 3150, 3150,2100,210,12600, 2520, 4200,120, 4725,  9450, 12600, 1260, 2520, 1575, 45, 12600, 3780, 12600, 360, 2800,  840, 1260, 10,  945,  3150,  6300,  630,  2520,  1575,  45,  2100, 120,  210, 126,  1,
1, 55,165, 990,330,4620,462, 6930, 6930,4620,462,34650, 6930,11550,330,17325, 34650, 46200, 4620, 9240, 5775,165, 69300,20790, 69300,1980,15400, 4620, 6930, 55,10395, 34650, 69300, 6930, 27720, 17325, 495, 23100,1320, 2310,1386, 11, 17325, 6930, 34650,  990, 15400, 4620, 6930, 55, 4620, 5775, 165, 330, 462,  1,
1, 66,220,1485,495,7920,792,13860,13860,9240,924,83160,16632,27720,792,51975,103950,138600,13860,27720,17325,495,277200,83160,277200,7920,61600,18480,27720,220,62370,207900,415800,41580,166320,103950,2970,138600,7920,13860,8316, 66,207900,83160,415800,11880,184800,55440,83160,660,55440,69300,1980,3960,5544, 12,10395,51975,138600,13860,83160,51975,1485,138600,7920,13860,8316, 66,15400,9240,27720,220,5775,495,792,462,  1

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All, Noncrossing        (by type)        Triangle A211351

0   1   2   3   4    5   6    7    8    9  10    11   12   13  14   15    16    17   18   19   20  21    22   23    24   25   26   27   28  29   30   31    32   33   34   35  36   37  38  39  40  41   42   43   44  45   46   47   48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69   70  71  72  73  74  75  76

1,  1,
1,  3,  1,
1,  6,  4,  2,  1,
1, 10, 10, 10,  5,   5,  1,
1, 15, 20, 30, 15,  30,  6,   5,   6,   3,  1,
1, 21, 35, 70, 35, 105, 21,  35,  42,  21,  7,   21,   7,   7,  1,
1, 28, 56,140, 70, 280, 56, 140, 168,  84, 28,  168,  56,  56,  8,  14,   28,   28,   8,   8,   4,  1,
1, 36, 84,252,126, 630,126, 420, 504, 252, 84,  756, 252, 252, 36, 126,  252,  252,  72,  72,  36,  9,   84,  36,   72,   9,  12,   9,   9,  1,
1, 45,120,420,210,1260,252,1050,1260, 630,210, 2520, 840, 840,120, 630, 1260, 1260, 360, 360, 180, 45,  840, 360,  720,  90, 120,  90,  90, 10,  42, 120,  180,  45,  90,  45, 10,  45, 10, 10,  5,  1,
1, 55,165,660,330,2310,462,2310,2772,1386,462, 6930,2310,2310,330,2310, 4620, 4620,1320,1320, 660,165, 4620,1980, 3960, 495, 660, 495, 495, 55, 462,1320, 1980, 495, 990, 495,110, 495,110,110, 55, 11, 330, 165, 495, 55, 165, 110, 110, 11, 55, 55, 11, 11, 11,  1,
1, 66,220,990,495,3960,792,4620,5544,2772,924,16632,5544,5544,792,6930,13860,13860,3960,3960,1980,495,18480,7920,15840,1980,2640,1980,1980,220,2772,7920,11880,2970,5940,2970,660,2970,660,660,330, 66,3960,1980,5940,660,1980,1320,1320,132,660,660,132,132,132, 12,132,495,990,220,660,330, 66,660,132,132, 66, 12, 55, 66, 132, 12, 22, 12, 12,  6,  1

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Rightmost columns: Pascal's triangle (A007318)

0       1   2   4   6  10  14  21  29  41  55  76

1,  1,
1,  2,  1,
1,  3,  3,  1,
1,  4,  6,  4,  1,
1,  5, 10, 10,  5,  1,
1,  6, 15, 20, 15,  6,  1,
1,  7, 21, 35, 35, 21,  7,  1,
1,  8, 28, 56, 70, 56, 28,  8,  1,
1,  9, 36, 84,126,126, 84, 36,  9,  1,
1, 10, 45,120,210,252,210,120, 45, 10,  1,
1, 11, 55,165,330,462,462,330,165, 55, 11,  1,
1, 12, 66,220,495,792,924,792,495,220, 66, 12,  1

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### Up to rotation

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Up to rot, All        (by number of singletons)
Triangle A211356 with row sums A084423

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |  | sum

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0  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
1  | 0 | 1 |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  | 3
4  | 3 | 1 | 2 | 0 | 1 |  |  |  |  |  |  |  |  | 7
5  | 3 | 4 | 2 | 2 | 0 | 1 |  |  |  |  |  |  |  | 12
6  | 12 | 11 | 12 | 4 | 3 | 0 | 1 |  |  |  |  |  |  | 43
7  | 24 | 41 | 33 | 20 | 5 | 3 | 0 | 1 |  |  |  |  |  | 127
8  | 103 | 162 | 151 | 77 | 39 | 7 | 4 | 0 | 1 |  |  |  |  | 544
9  | 387 | 715 | 648 | 386 | 154 | 56 | 10 | 4 | 0 | 1 |  |  |  | 2361
10  | 1819 | 3425 | 3255 | 1944 | 876 | 278 | 88 | 12 | 5 | 0 | 1 |  |  | 11703
11  | 8933 | 17722 | 17125 | 10725 | 4860 | 1722 | 462 | 120 | 15 | 5 | 0 | 1 |  | 61690
12  | 48632 | 98253 | 97685 | 62807 | 29593 | 10692 | 3188 | 726 | 171 | 19 | 6 | 0 | 1 | 351773

Up to rot, Noncrossing        (by number of singletons)
Triangle A211357 with row sums A054357

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |  | sum

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0  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
1  | 0 | 1 |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  | 3
4  | 2 | 1 | 2 | 0 | 1 |  |  |  |  |  |  |  |  | 6
5  | 2 | 3 | 2 | 2 | 0 | 1 |  |  |  |  |  |  |  | 10
6  | 5 | 6 | 9 | 4 | 3 | 0 | 1 |  |  |  |  |  |  | 28
7  | 6 | 15 | 18 | 15 | 5 | 3 | 0 | 1 |  |  |  |  |  | 63
8  | 15 | 36 | 56 | 42 | 29 | 7 | 4 | 0 | 1 |  |  |  |  | 190
9  | 28 | 91 | 144 | 142 | 84 | 42 | 10 | 4 | 0 | 1 |  |  |  | 546
10  | 67 | 232 | 419 | 432 | 322 | 152 | 66 | 12 | 5 | 0 | 1 |  |  | 1708
11  | 145 | 603 | 1160 | 1365 | 1080 | 630 | 252 | 90 | 15 | 5 | 0 | 1 |  | 5346
12  | 368 | 1585 | 3342 | 4258 | 3779 | 2376 | 1170 | 396 | 128 | 19 | 6 | 0 | 1 | 17428

By type

Up to rot, All        (by type)        Triangle A211352

0   1   2   3   4   5   6    7    8   9  10   11   12   13  14   15   16    17   18   19   20  21    22   23    24  25   26   27   28  29   30    31    32   33    34   35  36    37  38   39  40  41    42   43    44  45    46   47   48  49   50   51  52  53  54  55  56   57    58   59   60   61  62    63  64   65  66  67   68  69   70  71  72  73  74  75  76

1,  1,
1,  1,  1,
1,  2,  1,  2,  1,
1,  2,  2,  3,  1,  2,  1,
1,  3,  4,  9,  3, 10,  1,   5,   3,  3,  1,
1,  3,  5, 15,  5, 30,  3,  15,  15, 10,  1,  15,   3,   5,  1,
1,  4,  7, 29, 10, 70,  7,  56,  54, 37,  4, 105,  21,  35,  1,  18,  29,   37,   4,   7,   7,  1,
1,  4, 10, 42, 14,140, 14, 142, 140, 94, 10, 420,  84, 140,  4, 105, 210,  280,  28,  56,  35,  1,  142,  42,  140,  4,  34,  10,  14,  1,
1,  5, 12, 66, 22,252, 26, 322, 318,214, 22,1260, 252, 420, 12, 485, 954, 1268, 128, 252, 163,  5, 1260, 378, 1260, 36, 280,  84, 126,  1, 105,  322,  642,  66,  252, 163,  5,  214, 12,  22, 15,  1,
1,  5, 15, 90, 30,420, 42, 630, 630,420, 42,3150, 630,1050, 30,1575,3150, 4200, 420, 840, 525, 15, 6300,1890, 6300,180,1400, 420, 630,  5, 945, 3150, 6300, 630, 2520,1575, 45, 2100,120, 210,126,  1, 1575, 630, 3150, 90, 1400, 420, 630,  5, 420, 525, 15, 30, 42,  1
1,  6, 19,128, 43,660, 66,1170,1160,778, 80,6930,1386,2310, 66,4365,8686,11570,1160,2310,1459, 43,23106,6930,23100,660,5140,1542,2310, 19,5238,17360,34710,3480,13860,8690,250,11570,660,1160,701,  6,17325,6930,34650,990,15400,4620,6930, 55,4620,5775,165,330,462,  1,902,4365,11600,1170,6930,4374,128,11570,660,1160,701,  6,1306,778,2310, 19,499, 43, 66, 44,  1

|

Up to rot, Noncrossing        (by type)        Triangle A211353

0   1   2   3   4   5   6   7   8   9  10   11  12  13  14  15   16   17  18  19  20  21   22  23   24  25  26  27  28  29  30  31   32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69  70  71  72  73  74  75  76

1,  1,
1,  1,  1,
1,  2,  1,  1,  1,
1,  2,  2,  2,  1,  1,  1,
1,  3,  4,  6,  3,  5,  1,  2,  1,  1,  1,
1,  3,  5, 10,  5, 15,  3,  5,  6,  3,  1,   3,  1,  1,  1,
1,  4,  7, 19, 10, 35,  7, 19, 21, 12,  4,  21,  7,  7,  1,  3,   4,   4,  1,  1,  1,  1,
1,  4, 10, 28, 14, 70, 14, 48, 56, 28, 10,  84, 28, 28,  4, 14,  28,  28,  8,  8,  4,  1,  10,  4,   8,  1,  2,  1,  1,  1,
1,  5, 12, 44, 22,126, 26,108,126, 66, 22, 252, 84, 84, 12, 66, 128, 128, 36, 36, 20,  5,  84, 36,  72,  9, 12,  9,  9,  1,  6, 12,  20,  5,  9,  5,  1,  5,  1,  1,  1,  1,
1,  5, 15, 60, 30,210, 42,210,252,126, 42, 630,210,210, 30,210, 420, 420,120,120, 60, 15, 420,180, 360, 45, 60, 45, 45,  5, 42,120, 180, 45, 90, 45, 10, 45, 10, 10,  5,  1, 30, 15, 45,  5, 15, 10, 10,  1,  5,  5,  1,  1,  1,  1,
1,  6, 19, 85, 43,330, 66,392,462,236, 80,1386,462,462, 66,586,1160,1160,330,330,170, 43,1542,660,1320,165,222,165,165, 19,236,660,1000,250,495,250, 55,250, 55, 55, 30,  6,330,165,495, 55,165,110,110, 11, 55, 55, 11, 11, 11,  1, 14, 43, 85, 19, 55, 30,  6, 55, 11, 11,  6,  1,  7,  6, 11,  1,  3,  1,  1,  1,  1

|

Rightmost columns: Triangle of circular binomial coefficients (A047996)

0       1   2   4   6  10  14  21  29  41  55  76

1,  1,
1,  1,  1,
1,  1,  1,  1,
1,  1,  2,  1,  1,
1,  1,  2,  2,  1,  1,
1,  1,  3,  4,  3,  1,  1,
1,  1,  3,  5,  5,  3,  1,  1,
1,  1,  4,  7, 10,  7,  4,  1,  1,
1,  1,  4, 10, 14, 14, 10,  4,  1,  1,
1,  1,  5, 12, 22, 26, 22, 12,  5,  1,  1,
1,  1,  5, 15, 30, 42, 42, 30, 15,  5,  1,  1,
1,  1,  6, 19, 43, 66, 80, 66, 43, 19,  6,  1,  1

|

The diagonal is A003239. According to the OEIS page this is also the number of necklaces with n white and n black beads.
Such necklaces correspond to integer partitions 0, 3, 9, 20, 40, 75, 133, 229, 383, 625, 1000... (compare). These numbers are found in the partition numbers minus 2:
A000041 = 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002...

### Up to rotation and reflection

[edit | edit source]

Up to rot & ref, All        (by number of singletons)
Triangle A211358 with row sums A084708

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |  | sum

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0  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
1  | 0 | 1 |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  | 3
4  | 3 | 1 | 2 | 0 | 1 |  |  |  |  |  |  |  |  | 7
5  | 3 | 4 | 2 | 2 | 0 | 1 |  |  |  |  |  |  |  | 12
6  | 12 | 7 | 11 | 3 | 3 | 0 | 1 |  |  |  |  |  |  | 37
7  | 20 | 28 | 21 | 16 | 4 | 3 | 0 | 1 |  |  |  |  |  | 93
8  | 81 | 89 | 101 | 43 | 30 | 5 | 4 | 0 | 1 |  |  |  |  | 354
9  | 238 | 395 | 356 | 223 | 86 | 40 | 7 | 4 | 0 | 1 |  |  |  | 1350
10  | 1079 | 1757 | 1783 | 1004 | 504 | 148 | 62 | 8 | 5 | 0 | 1 |  |  | 6351
11  | 4752 | 9075 | 8785 | 5550 | 2510 | 936 | 246 | 80 | 10 | 5 | 0 | 1 |  | 31950
12  | 25421 | 49412 | 49904 | 31626 | 15279 | 5426 | 1729 | 378 | 113 | 12 | 6 | 0 | 1 | 179307

Up to rot & ref, Noncrossing        (by number of singletons)
Triangle A211359 with row sums A111275

|  | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |  | sum

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0  | 1 |  |  |  |  |  |  |  |  |  |  |  |  | 1
1  | 0 | 1 |  |  |  |  |  |  |  |  |  |  |  | 1
2  | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  |  | 2
3  | 1 | 1 | 0 | 1 |  |  |  |  |  |  |  |  |  | 3
4  | 2 | 1 | 2 | 0 | 1 |  |  |  |  |  |  |  |  | 6
5  | 2 | 3 | 2 | 2 | 0 | 1 |  |  |  |  |  |  |  | 10
6  | 5 | 4 | 8 | 3 | 3 | 0 | 1 |  |  |  |  |  |  | 24
7  | 6 | 11 | 12 | 12 | 4 | 3 | 0 | 1 |  |  |  |  |  | 49
8  | 14 | 21 | 39 | 24 | 22 | 5 | 4 | 0 | 1 |  |  |  |  | 130
9  | 22 | 55 | 84 | 85 | 48 | 30 | 7 | 4 | 0 | 1 |  |  |  | 336
10  | 51 | 124 | 245 | 228 | 190 | 82 | 46 | 8 | 5 | 0 | 1 |  |  | 980
11  | 95 | 327 | 620 | 730 | 570 | 350 | 136 | 60 | 10 | 5 | 0 | 1 |  | 2904
12  | 232 | 815 | 1784 | 2169 | 2002 | 1218 | 645 | 208 | 84 | 12 | 6 | 0 | 1 | 9176

By type

Up to rot & ref, All        (by type)        Triangle A211354

0   1   2   3   4   5   6   7   8   9  10   11  12   13  14   15   16   17  18   19  20  21    22   23    24  25   26  27   28  29   30   31    32   33   34   35  36   37  38  39  40  41   42   43    44  45   46   47   48  49   50   51  52  53  54  55  56   57   58  59   60   61  62   63  64  65  66  67  68  69   70  71  72  73  74  75  76

1,  1,
1,  1,  1,
1,  2,  1,  2,  1,
1,  2,  2,  3,  1,  2,  1,
1,  3,  3,  8,  3,  6,  1,  5,  3,  3,  1,
1,  3,  4, 12,  4, 18,  3, 11,  9,  7,  1,  12,  3,   4,  1,
1,  4,  5, 22,  8, 38,  5, 39, 33, 25,  4,  57, 12,  19,  1,  17,  22,  25,  4,   5,  7,  1,
1,  4,  7, 30, 10, 76, 10, 85, 76, 55,  7, 228, 48,  76,  4,  65, 114, 148, 16,  28, 23,  1,   85,  30,   76,  4,  25,  7,  10,  1,
1,  5,  8, 46, 16,132, 16,190,174,124, 16, 648,132, 216,  8, 287, 516, 668, 74, 132,101,  5,  644, 198,  636, 20, 148, 44,  66,  1,  79, 190,  364,  46, 132, 101,  5, 124,  8, 16, 13,  1,
1,  5, 10, 60, 20,220, 26,350,330,230, 26,1620,330, 540, 20, 850,1620,2140,220, 420,290, 10, 3220, 990, 3180,100, 740,220, 330,  5, 513,1610, 3210, 330,1260, 815, 25,1070, 60,110, 71,  1, 850, 350, 1620, 60, 740, 220, 330,  5, 230, 290, 10, 20, 26,  1
1,  6, 12, 84, 29,340, 38,645,610,424, 50,3510,708,1170, 38,2325,4463,5890,610,1170,792, 29,11623,3510,11580,340,2610,781,1170, 12,2792,8860,17630,1820,6960,4470,140,5890,340,610,386,  6,8725,3500,17370,510,7740,2320,3480, 30,2330,2915, 85,170,236,  1,554,2325,6005,645,3510,2347, 84,5890,340,610,386,  6,713,424,1170, 12,297, 29, 38, 35,  1

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Up to rot & ref, Noncrossing        (by type)        Triangle A211355

0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69  70  71  72  73  74  75  76

1,  1,
1,  1,  1,
1,  2,  1,  1,  1,
1,  2,  2,  2,  1,  1,  1,
1,  3,  3,  5,  3,  3,  1,  2,  1,  1,  1,
1,  3,  4,  8,  4,  9,  3,  4,  4,  2,  1,  3,  1,  1,  1,
1,  4,  5, 14,  8, 19,  5, 14, 13,  8,  4, 12,  4,  4,  1,  3,  4,  3,  1,  1,  1,  1,
1,  4,  7, 20, 10, 38, 10, 30, 32, 16,  7, 48, 16, 16,  4, 10, 17, 15,  5,  4,  3,  1,  7,  4,  5,  1,  2,  1,  1,  1,
1,  5,  8, 30, 16, 66, 16, 66, 70, 38, 16,132, 44, 44,  8, 42, 75, 69, 21, 20, 13,  5, 44, 20, 37,  5,  7,  5,  5,  1,  6,  9, 13,  5,  5,  4,  1,  4,  1,  1,  1,  1,
1,  5, 10, 40, 20,110, 26,120,136, 68, 26,330,110,110, 20,120,225,215, 65, 60, 35, 10,220,100,185, 25, 35, 25, 25,  5, 26, 66, 94, 26, 45, 25,  6, 24,  5,  6,  3,  1, 20, 10, 26,  5,  9,  6,  6,  1,  4,  4,  1,  1,  1,  1,
1,  6, 12, 55, 29,170, 38,221,246,128, 50,708,236,236, 38,323,615,595,175,170, 95, 29,781,340,665, 85,116, 85, 85, 12,138,351,519,141,250,135, 31,134, 30, 31, 18,  6,170, 85,251, 30, 84, 56, 56,  6, 29, 29,  6,  6,  6,  1, 12, 30, 49, 13, 30, 20,  6, 30,  6,  7,  4,  1,  5,  5,  6,  1,  3,  1,  1,  1,  1

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Rightmost columns: A052307

0       1   2   4   6  10  14  21  29  41  55  76

1,  1,
1,  1,  1,
1,  1,  1,  1,
1,  1,  2,  1,  1,
1,  1,  2,  2,  1,  1,
1,  1,  3,  3,  3,  1,  1,
1,  1,  3,  4,  4,  3,  1,  1,
1,  1,  4,  5,  8,  5,  4,  1,  1,
1,  1,  4,  7, 10, 10,  7,  4,  1,  1,
1,  1,  5,  8, 16, 16, 16,  8,  5,  1,  1,
1,  1,  5, 10, 20, 26, 26, 20, 10,  5,  1,  1,
1,  1,  6, 12, 29, 38, 50, 38, 29, 12,  6,  1,  1

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### Subsequences

[edit | edit source]

The by type triangles have some interesting subsequences which are highlighted by colors in the files linked below.

Color  | Size  | Integer partition

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|

■ | 2n | 1 * n    \+    n * 1
■ | 2n | n * 2
□ | 2n | 2 * n
■ | 2n + 1 | 1 * n    \+    (n+1) * 1
■ | 2n + 1 | n * 1    \+    1 * (n+1)
□ | 2n + 1 | 1 * n    \+    1 * (n+1)

1 * n \+ n * 1 stands for the integer partition with one addend n and n addends 1, etc.

| All  | Noncrossing

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All  |

Triangle A211350

| ■ | 1,6,20,70,252,924...
(2,6,20... are the central binomial coefficients A000984)

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■ | 1,3,15,105,945,10395...     (double factorial numbers A001147)

■ □ | 1,3,10,35,126,462...     (A001700)
1,10,35,126,462...
3,10,35,126,462...     (A001700)

Triangle A211351

| ■ | 1,6,20,70,252,924...
(2,6,20... are the central binomial coefficients A000984)

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■ | 1,2,5,14,42,132...     (Catalan numbers A000108)
□ | 1,2,3,4,5,6...     (the natural numbers)

■ | 1,10,35,126,462...
3,10,35,126,462...     (A001700)
□ | 3,5,7,9,11     (odd numbers)

rot  |

Triangle A211352

| ■ | 1,2,4,10,26,80...     (A003239)

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■ | 1,2,5,18,105,902...     (A007769)
□ | 1,2,3,7,15,44...     (A045629)
■ ■ □ | 1,2,5,14,42...     (Catalan numbers A000108)

Triangle A211353

| ■ | 1,2,4,10,26,80...     (A003239)

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■ | 1,1,2,3,6,14...     (A002995)
□ □ | 1,1,1,1,1...
■ ■ | 1,2,5,14,42...     (Catalan numbers A000108)

rot

ref  |

Triangle A211354

| ■ | 1,2,3,8,16,50...     (A005648)

|

■ | 1,2,5,17,79,554...     (A054499)
□ | 1,2,3,7,13,35...     (A006840?)
■ ■ □ | 1,2,4,10,26...     (A003239)

Triangle A211355

| ■ | 1,2,3,8,16,50...     (A005648)

|

■ | 1,1,2,3,6,12...
□ □ | 1,1,1,1,1...
■ ■ | 1,2,4,10,26...     (A003239)

N   =  0, 1, 2,  3,  4,   5,   6...
A000984(N)  =  1, 2, 6, 20, 70, 252, 924...          (central binomial coefficients)

* The sequence 1,6,20,70,252,924... appears as All, All ■ and All, Noncrossing ■ :

There is one partition of a 2-set that has a 1-block (i.e. a singleton) and another singleton.
For n>1 there are A000984(n) partitions of a 2n-set that have an n-block and n singletons.

N   =  0, 1,  2,  3,   4,   5...
A001700(N)  =  1, 3, 10, 35, 126, 462...

* All, All □ :

There are A001700(n+1) partitions of a 2n-set that have two n-blocks.

* All, All ■ and All, Noncrossing  ■ :

There is one partition of a 3-set that has a 1-block (i.e. a singleton) and two other singletons.
For n>1 there are A001700(n) partitions of a (2n+1)-set that have an n-block and n+1 singletons.

* All, All ■ and All, Noncrossing ■ :

There are A001700(n) partitions of a (2n+1)-set that have an (n+1)-block and n singletons.

* All, All □ :

There are A001700(n) partitions of a (2n+1)-set that have an (n+1)-block and an n-block.

N+   =  1, 2, 3,  4,  5,  6...
A003239(N+)  =  1, 2, 4, 10, 26, 80...

* Up to rot, All ■ and Up to rot, Noncrossing ■ :

There are A003239(n) necklaces of length 2n with n black and n white beads.

* Up to rot & ref, All ■ ■ □ and Up to rot & ref, Noncrossing ■ ■ :

There are A003239(n) free necklaces (or bracelets) of length 2n+1 with n black and n+1 white beads.
*[A]: All
*[N]: Only noncrossing
*[Utr]: Up to rotation
*[Utr&r]: Up to rotation and reflection
*[Rc]: Rightmost columns