# Partition related number triangles
## Set partitions by number of blocks

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 |
| All | Triangle A008277 (Stirling2) with row sums A000110 (Bell) 12345678910111213sum 1 11 2 112 3 1315 4 176115 5 1152510152 6 1319065151203 7 163301350140211877 8 1127966170110502662814140 9 1255302577706951264646236121147 10 151193303410542525228275880750451115975 11 110232850114575024673017948763987118801155551678570 12 1204786526611501137940013236526273961590272227517056614213597 13 140952616252532530750850193213125715424189961235950239325243178127644437 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) 12345678910111213sum 1 11 2 112 3 1315 4 166114 5 1102010142 6 1155050151132 7 121105175105211429 8 1281964904901962811430 9 1363361176176411763363614862 10 145540252052925292252054045116796 11 1558254950138601940413860495082555158786 12 166121090753267060984609843267090751210661208012 13 1781716157307078516988422651216988470785157301716781742900 Diagonal: A000891 The 1,6,6,1 n. p. of a 4-element set with 1,2,3,4 blocks |
| Up to rot | Triangle A152175 with row sums A084423 1234567891011sum 1 11 2 112 3 1113 4 13217 5 1352112 6 1718133143 7 1943502031127 8 1191262211363641544 9 12933986677329652412361 10 155946343742812303596785111703 11 1932591132502243016317581710801055161690 | Triangle A209805 with row sums A054357 12345678910111213sum 1 11 2 112 3 1113 4 12216 5 1242110 6 1310103128 7 131525153163 8 142664642641190 9 14381321961323841546 10 155625653653625656511708 11 157545012601764126045075515346 12 1610476527365102510227367651046117428 13 1613212105445130681742413068544512101326157148 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. |
| Up to rot & ref | Triangle A152176 with row sums A084708 1234567891011sum 1 11 2 112 3 1113 4 13217 5 1352112 6 1714113137 7 183133163193 8 11782137852741354 9 12220247843417137411350 10 14353818512271124933854516351 11 162140168451153083893056590705131950 | Triangle A209612 with row sums A111275 1234567891011sum 1 11 2 112 3 1113 4 12216 5 1242110 6 13883124 7 131217123149 8 141941411941130 9 142778116782741336 10 15381482982981483851980 11 155025068093268025050512904 |
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:
| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | sum |
| 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 |

### Lah numbers

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).

## Set partitions by type and by number of singletons

| tabl triangles, columns match number of singletons: AN A A124323 A091867 Utr A211356 A211357 Utr&r A211358 A211359 tabf triangles, columns match type: AN   Rc A A211350 A211351 A007318 Utr A211352 A211353 A047996 Utr&r A211354 A211355 A052307 Sequences of row sums: AN 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

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | sum |
| 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 |
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | sum |
| 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 :
| 1 2 3 4 5 6 7 8 9 10 11 12 | 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, 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 |
| 1 2 3 4 5 6 7 8 9 10 11 12 | 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, 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 |
| 1 2 3 4 5 6 7 8 9 10 11 12 | 0 1 2 4 6 10 14 21 29 41 55 76 1, 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 |

### Up to rotation

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | sum |
| 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 |
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | sum |
| 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 :
| 1 2 3 4 5 6 7 8 9 10 11 12 | 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, 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 |
| 1 2 3 4 5 6 7 8 9 10 11 12 | 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, 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 |
| 1 2 3 4 5 6 7 8 9 10 11 12 | 0 1 2 4 6 10 14 21 29 41 55 76 1, 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 |

### Up to rotation and reflection

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | sum |
| 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 |
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | sum |
| 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 :
| 1 2 3 4 5 6 7 8 9 10 11 12 | 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, 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 |
| 1 2 3 4 5 6 7 8 9 10 11 12 | 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, 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 |
| 1 2 3 4 5 6 7 8 9 10 11 12 | 0 1 2 4 6 10 14 21 29 41 55 76 1, 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 |

### Subsequences

The by type triangles have some interesting subsequences which are highlighted by colors in the files linked below.
| Color | Size | Integer partition |
| ■ | 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 |
| All | Triangle A211350   ■1,6,20,70,252,924... (2,6,20... are the central binomial coefficients A000984) ■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) ■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) |
| Up to rot | Triangle A211352   ■1,2,4,10,26,80... (A003239) ■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) ■1,1,2,3,6,14... (A002995) □ □1,1,1,1,1... ■ ■1,2,5,14,42... (Catalan numbers A000108) |
| Up to 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) |
* 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.
* 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.
* 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.