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# Hand Distribution Table

I am including this information in my web pages because I think it is neat and that it is a method I think other writers of hand generators might want to use.

It is also the only time I can think of when I have used anything I learned in graduate school.

## Creation of the table

My program, 'Deal', needed a fast way to determine whether a hand was in a particular class. The goal was to stay out of the TCL interpreter as much as possible. The answer was to implement a lookup table with an easy indexing algorithm.

There are 560 hand shapes, where by "shape", I mean the ordered listing of suit lengths: spades-hearts-diamonds-clubs.

The hand shapes are in easy 1-1 correspondence with the 3-subsets of {0,...,15}. In particular, the hand shape with s spades, h hearts, d diamonds, and c clubs corresponds to the 3-set {s, s+h+1, s+h+d+2}.

There is a linear order on n-subsets, for fixed n, called the squashed order [see Combinatorics on Finite Sets, Anderson, pp 112-119.] The nice thing about this order is that it is easy to find the index of an n-set in the order. In the case of n=3, take a subset {x,y,z}, with 0<=x<y<z<=15. The index of this set in the squashed order is

(z choose 3) + (y choose 2) + (x choose 1)
For the hand shape s-h-d-c, then, the corresponding index in the squashed order is
(s+h+d+2 choose 3) + (s+h+1 choose 2) + (s choose 1)

For added speed, I pre-computed (n+2 choose 3) and (n+1 choose 2) for n=0,...13, placing the values in static arrays. The resulting C code looks like:

```static int distTableIndex(s,h,d)
int s,h,d;
{
static choose2tab[]={0, 1, 3,  6, 10, 15, 21, 28,  36,  45,  55,  66,  78,  91};
static choose3tab[]={0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455};
return choose3tab[s+h+d]+choose2tab[s+h]+s;
}
```
It is precisely because I can quickly compute this index that I have chosen the squashed order. Perhaps other orders also happen to allow for quick indexing, but this is the one I found in my personal math library.

## Usage of the table

The table is useful for fast lookups. For example, lets say you were looking for hands which were five-five or better in any two suits. You could, of course, simply check each suit length, but the problem with this is that in an interpreted language, like Tcl, that might be too slow. If, however, you built a 560 element binary array, and do the computation 560 times up front, then when we start analyzing actual deals, we can quickly determine whether a particular shape was in that class.

In reality, the table is not computed until the first request is made of a shapeclass, which allows users to create large libraries to be included but not instantiated until used.

For example:

```shapecond Balanced {\$s*\$s+\$h*\$h+\$d*\$d+\$c*\$c<=47}
```
When this shapeclass is first used, a table of 560 entries is created, and then the interpreter evaluates this expression only 560 times. For rare hand classes, this is a significant improvement.

When Deal is confronted with a specific query about membership in this class, all it has to do is use distTableIndex find the index, then look it up in the table. This is significantly faster than reinterpreting the Tcl code every time.

The concept of "shape class" led to the concept of "shape function". These are functions which use the shape of the hand to determine the value. For instance, I have a function which determines the opening suit for a hand:

```shapefunc opensuit {

if {\$s>=5 && \$s>=\$h && \$s>=\$d && \$s>=\$c} {return spades}

if {\$h>=5 && \$h>=\$d && \$h>=\$c} { return hearts }

if {\$d>=4 && \$d>=\$c} {return diamonds}

if {\$c<3} {return diamonds}

return clubs
}
```
[ The current implementation of shapefunc is a bit of a memory hog, unfortunately, since it allocates strings for every element of the table, even if the strings are duplicates. For instance, the function above has only 4 return values, "spades", "hearts", "diamonds", and "clubs". Still, shapefunc instantiates 560 strings. Smarter implementations are certainly possible, and Tcl 8.0 alleviates the problem considerably (because it allows deals with reference-counted strings.) ]

[ We could also instantiate the table an entry at a time, leaving null pointers in the table until a value has been requested. This has the advantage that we will often compute considerably fewer values from the table. The disadvantage is that every time we need a value from the table, we will have to do a check to see if a pointer value is null. It is not clear to me this would be an improvement or not, but it would add a complexity to the code that I am not willing to maintain. It might seem that you would need to check a pointer anyway, to determine if the shapeclass has been instantiated or not, but the implementation avoids such a check.]

### Note

The above definition of Balanced is an interesting oddity, which I leave it up to the reader to try to understand. Think about it for a moment before you look here.

## Table

Here is the raw data of the table. Not very interesting, but it does help to make it clear how the squashed ordering works, and why the index values are computed as they are. Index| S H D C ================== 0 | 0 0 0 13 1 | 0 0 1 12 2 | 0 1 0 12 3 | 1 0 0 12 4 | 0 0 2 11 5 | 0 1 1 11 6 | 1 0 1 11 7 | 0 2 0 11 8 | 1 1 0 11 9 | 2 0 0 11 10 | 0 0 3 10 11 | 0 1 2 10 12 | 1 0 2 10 13 | 0 2 1 10 14 | 1 1 1 10 15 | 2 0 1 10 16 | 0 3 0 10 17 | 1 2 0 10 18 | 2 1 0 10 19 | 3 0 0 10 20 | 0 0 4 9 21 | 0 1 3 9 22 | 1 0 3 9 23 | 0 2 2 9 24 | 1 1 2 9 25 | 2 0 2 9 26 | 0 3 1 9 27 | 1 2 1 9 28 | 2 1 1 9 29 | 3 0 1 9 30 | 0 4 0 9 31 | 1 3 0 9 32 | 2 2 0 9 33 | 3 1 0 9 34 | 4 0 0 9 35 | 0 0 5 8 36 | 0 1 4 8 37 | 1 0 4 8 38 | 0 2 3 8 39 | 1 1 3 8 40 | 2 0 3 8 41 | 0 3 2 8 42 | 1 2 2 8 43 | 2 1 2 8 44 | 3 0 2 8 45 | 0 4 1 8 46 | 1 3 1 8 47 | 2 2 1 8 48 | 3 1 1 8 49 | 4 0 1 8 50 | 0 5 0 8 51 | 1 4 0 8 52 | 2 3 0 8 53 | 3 2 0 8 54 | 4 1 0 8 55 | 5 0 0 8 56 | 0 0 6 7 57 | 0 1 5 7 58 | 1 0 5 7 59 | 0 2 4 7 60 | 1 1 4 7 61 | 2 0 4 7 62 | 0 3 3 7 63 | 1 2 3 7 64 | 2 1 3 7 65 | 3 0 3 7 66 | 0 4 2 7 67 | 1 3 2 7 68 | 2 2 2 7 69 | 3 1 2 7 70 | 4 0 2 7 71 | 0 5 1 7 72 | 1 4 1 7 73 | 2 3 1 7 74 | 3 2 1 7 75 | 4 1 1 7 76 | 5 0 1 7 77 | 0 6 0 7 78 | 1 5 0 7 79 | 2 4 0 7 80 | 3 3 0 7 81 | 4 2 0 7 82 | 5 1 0 7 83 | 6 0 0 7 84 | 0 0 7 6 85 | 0 1 6 6 86 | 1 0 6 6 87 | 0 2 5 6 88 | 1 1 5 6 89 | 2 0 5 6 90 | 0 3 4 6 91 | 1 2 4 6 92 | 2 1 4 6 93 | 3 0 4 6 94 | 0 4 3 6 95 | 1 3 3 6 96 | 2 2 3 6 97 | 3 1 3 6 98 | 4 0 3 6 99 | 0 5 2 6 100 | 1 4 2 6 101 | 2 3 2 6 102 | 3 2 2 6 103 | 4 1 2 6 104 | 5 0 2 6 105 | 0 6 1 6 106 | 1 5 1 6 107 | 2 4 1 6 108 | 3 3 1 6 109 | 4 2 1 6 110 | 5 1 1 6 111 | 6 0 1 6 112 | 0 7 0 6 113 | 1 6 0 6 114 | 2 5 0 6 115 | 3 4 0 6 116 | 4 3 0 6 117 | 5 2 0 6 118 | 6 1 0 6 119 | 7 0 0 6 120 | 0 0 8 5 121 | 0 1 7 5 122 | 1 0 7 5 123 | 0 2 6 5 124 | 1 1 6 5 125 | 2 0 6 5 126 | 0 3 5 5 127 | 1 2 5 5 128 | 2 1 5 5 129 | 3 0 5 5 130 | 0 4 4 5 131 | 1 3 4 5 132 | 2 2 4 5 133 | 3 1 4 5 134 | 4 0 4 5 135 | 0 5 3 5 136 | 1 4 3 5 137 | 2 3 3 5 138 | 3 2 3 5 139 | 4 1 3 5 140 | 5 0 3 5 141 | 0 6 2 5 142 | 1 5 2 5 143 | 2 4 2 5 144 | 3 3 2 5 145 | 4 2 2 5 146 | 5 1 2 5 147 | 6 0 2 5 148 | 0 7 1 5 149 | 1 6 1 5 150 | 2 5 1 5 151 | 3 4 1 5 152 | 4 3 1 5 153 | 5 2 1 5 154 | 6 1 1 5 155 | 7 0 1 5 156 | 0 8 0 5 157 | 1 7 0 5 158 | 2 6 0 5 159 | 3 5 0 5 160 | 4 4 0 5 161 | 5 3 0 5 162 | 6 2 0 5 163 | 7 1 0 5 164 | 8 0 0 5 165 | 0 0 9 4 166 | 0 1 8 4 167 | 1 0 8 4 168 | 0 2 7 4 169 | 1 1 7 4 170 | 2 0 7 4 171 | 0 3 6 4 172 | 1 2 6 4 173 | 2 1 6 4 174 | 3 0 6 4 175 | 0 4 5 4 176 | 1 3 5 4 177 | 2 2 5 4 178 | 3 1 5 4 179 | 4 0 5 4 180 | 0 5 4 4 181 | 1 4 4 4 182 | 2 3 4 4 183 | 3 2 4 4 184 | 4 1 4 4 185 | 5 0 4 4 186 | 0 6 3 4 187 | 1 5 3 4 188 | 2 4 3 4 189 | 3 3 3 4 190 | 4 2 3 4 191 | 5 1 3 4 192 | 6 0 3 4 193 | 0 7 2 4 194 | 1 6 2 4 195 | 2 5 2 4 196 | 3 4 2 4 197 | 4 3 2 4 198 | 5 2 2 4 199 | 6 1 2 4 200 | 7 0 2 4 201 | 0 8 1 4 202 | 1 7 1 4 203 | 2 6 1 4 204 | 3 5 1 4 205 | 4 4 1 4 206 | 5 3 1 4 207 | 6 2 1 4 208 | 7 1 1 4 209 | 8 0 1 4 210 | 0 9 0 4 211 | 1 8 0 4 212 | 2 7 0 4 213 | 3 6 0 4 214 | 4 5 0 4 215 | 5 4 0 4 216 | 6 3 0 4 217 | 7 2 0 4 218 | 8 1 0 4 219 | 9 0 0 4 220 | 0 0 10 3 221 | 0 1 9 3 222 | 1 0 9 3 223 | 0 2 8 3 224 | 1 1 8 3 225 | 2 0 8 3 226 | 0 3 7 3 227 | 1 2 7 3 228 | 2 1 7 3 229 | 3 0 7 3 230 | 0 4 6 3 231 | 1 3 6 3 232 | 2 2 6 3 233 | 3 1 6 3 234 | 4 0 6 3 235 | 0 5 5 3 236 | 1 4 5 3 237 | 2 3 5 3 238 | 3 2 5 3 239 | 4 1 5 3 240 | 5 0 5 3 241 | 0 6 4 3 242 | 1 5 4 3 243 | 2 4 4 3 244 | 3 3 4 3 245 | 4 2 4 3 246 | 5 1 4 3 247 | 6 0 4 3 248 | 0 7 3 3 249 | 1 6 3 3 250 | 2 5 3 3 251 | 3 4 3 3 252 | 4 3 3 3 253 | 5 2 3 3 254 | 6 1 3 3 255 | 7 0 3 3 256 | 0 8 2 3 257 | 1 7 2 3 258 | 2 6 2 3 259 | 3 5 2 3 260 | 4 4 2 3 261 | 5 3 2 3 262 | 6 2 2 3 263 | 7 1 2 3 264 | 8 0 2 3 265 | 0 9 1 3 266 | 1 8 1 3 267 | 2 7 1 3 268 | 3 6 1 3 269 | 4 5 1 3 270 | 5 4 1 3 271 | 6 3 1 3 272 | 7 2 1 3 273 | 8 1 1 3 274 | 9 0 1 3 275 | 0 10 0 3 276 | 1 9 0 3 277 | 2 8 0 3 278 | 3 7 0 3 279 | 4 6 0 3 280 | 5 5 0 3 281 | 6 4 0 3 282 | 7 3 0 3 283 | 8 2 0 3 284 | 9 1 0 3 285 |10 0 0 3 286 | 0 0 11 2 287 | 0 1 10 2 288 | 1 0 10 2 289 | 0 2 9 2 290 | 1 1 9 2 291 | 2 0 9 2 292 | 0 3 8 2 293 | 1 2 8 2 294 | 2 1 8 2 295 | 3 0 8 2 296 | 0 4 7 2 297 | 1 3 7 2 298 | 2 2 7 2 299 | 3 1 7 2 300 | 4 0 7 2 301 | 0 5 6 2 302 | 1 4 6 2 303 | 2 3 6 2 304 | 3 2 6 2 305 | 4 1 6 2 306 | 5 0 6 2 307 | 0 6 5 2 308 | 1 5 5 2 309 | 2 4 5 2 310 | 3 3 5 2 311 | 4 2 5 2 312 | 5 1 5 2 313 | 6 0 5 2 314 | 0 7 4 2 315 | 1 6 4 2 316 | 2 5 4 2 317 | 3 4 4 2 318 | 4 3 4 2 319 | 5 2 4 2 320 | 6 1 4 2 321 | 7 0 4 2 322 | 0 8 3 2 323 | 1 7 3 2 324 | 2 6 3 2 325 | 3 5 3 2 326 | 4 4 3 2 327 | 5 3 3 2 328 | 6 2 3 2 329 | 7 1 3 2 330 | 8 0 3 2 331 | 0 9 2 2 332 | 1 8 2 2 333 | 2 7 2 2 334 | 3 6 2 2 335 | 4 5 2 2 336 | 5 4 2 2 337 | 6 3 2 2 338 | 7 2 2 2 339 | 8 1 2 2 340 | 9 0 2 2 341 | 0 10 1 2 342 | 1 9 1 2 343 | 2 8 1 2 344 | 3 7 1 2 345 | 4 6 1 2 346 | 5 5 1 2 347 | 6 4 1 2 348 | 7 3 1 2 349 | 8 2 1 2 350 | 9 1 1 2 351 |10 0 1 2 352 | 0 11 0 2 353 | 1 10 0 2 354 | 2 9 0 2 355 | 3 8 0 2 356 | 4 7 0 2 357 | 5 6 0 2 358 | 6 5 0 2 359 | 7 4 0 2 360 | 8 3 0 2 361 | 9 2 0 2 362 |10 1 0 2 363 |11 0 0 2 364 | 0 0 12 1 365 | 0 1 11 1 366 | 1 0 11 1 367 | 0 2 10 1 368 | 1 1 10 1 369 | 2 0 10 1 370 | 0 3 9 1 371 | 1 2 9 1 372 | 2 1 9 1 373 | 3 0 9 1 374 | 0 4 8 1 375 | 1 3 8 1 376 | 2 2 8 1 377 | 3 1 8 1 378 | 4 0 8 1 379 | 0 5 7 1 380 | 1 4 7 1 381 | 2 3 7 1 382 | 3 2 7 1 383 | 4 1 7 1 384 | 5 0 7 1 385 | 0 6 6 1 386 | 1 5 6 1 387 | 2 4 6 1 388 | 3 3 6 1 389 | 4 2 6 1 390 | 5 1 6 1 391 | 6 0 6 1 392 | 0 7 5 1 393 | 1 6 5 1 394 | 2 5 5 1 395 | 3 4 5 1 396 | 4 3 5 1 397 | 5 2 5 1 398 | 6 1 5 1 399 | 7 0 5 1 400 | 0 8 4 1 401 | 1 7 4 1 402 | 2 6 4 1 403 | 3 5 4 1 404 | 4 4 4 1 405 | 5 3 4 1 406 | 6 2 4 1 407 | 7 1 4 1 408 | 8 0 4 1 409 | 0 9 3 1 410 | 1 8 3 1 411 | 2 7 3 1 412 | 3 6 3 1 413 | 4 5 3 1 414 | 5 4 3 1 415 | 6 3 3 1 416 | 7 2 3 1 417 | 8 1 3 1 418 | 9 0 3 1 419 | 0 10 2 1 420 | 1 9 2 1 421 | 2 8 2 1 422 | 3 7 2 1 423 | 4 6 2 1 424 | 5 5 2 1 425 | 6 4 2 1 426 | 7 3 2 1 427 | 8 2 2 1 428 | 9 1 2 1 429 |10 0 2 1 430 | 0 11 1 1 431 | 1 10 1 1 432 | 2 9 1 1 433 | 3 8 1 1 434 | 4 7 1 1 435 | 5 6 1 1 436 | 6 5 1 1 437 | 7 4 1 1 438 | 8 3 1 1 439 | 9 2 1 1 440 |10 1 1 1 441 |11 0 1 1 442 | 0 12 0 1 443 | 1 11 0 1 444 | 2 10 0 1 445 | 3 9 0 1 446 | 4 8 0 1 447 | 5 7 0 1 448 | 6 6 0 1 449 | 7 5 0 1 450 | 8 4 0 1 451 | 9 3 0 1 452 |10 2 0 1 453 |11 1 0 1 454 |12 0 0 1 455 | 0 0 13 0 456 | 0 1 12 0 457 | 1 0 12 0 458 | 0 2 11 0 459 | 1 1 11 0 460 | 2 0 11 0 461 | 0 3 10 0 462 | 1 2 10 0 463 | 2 1 10 0 464 | 3 0 10 0 465 | 0 4 9 0 466 | 1 3 9 0 467 | 2 2 9 0 468 | 3 1 9 0 469 | 4 0 9 0 470 | 0 5 8 0 471 | 1 4 8 0 472 | 2 3 8 0 473 | 3 2 8 0 474 | 4 1 8 0 475 | 5 0 8 0 476 | 0 6 7 0 477 | 1 5 7 0 478 | 2 4 7 0 479 | 3 3 7 0 480 | 4 2 7 0 481 | 5 1 7 0 482 | 6 0 7 0 483 | 0 7 6 0 484 | 1 6 6 0 485 | 2 5 6 0 486 | 3 4 6 0 487 | 4 3 6 0 488 | 5 2 6 0 489 | 6 1 6 0 490 | 7 0 6 0 491 | 0 8 5 0 492 | 1 7 5 0 493 | 2 6 5 0 494 | 3 5 5 0 495 | 4 4 5 0 496 | 5 3 5 0 497 | 6 2 5 0 498 | 7 1 5 0 499 | 8 0 5 0 500 | 0 9 4 0 501 | 1 8 4 0 502 | 2 7 4 0 503 | 3 6 4 0 504 | 4 5 4 0 505 | 5 4 4 0 506 | 6 3 4 0 507 | 7 2 4 0 508 | 8 1 4 0 509 | 9 0 4 0 510 | 0 10 3 0 511 | 1 9 3 0 512 | 2 8 3 0 513 | 3 7 3 0 514 | 4 6 3 0 515 | 5 5 3 0 516 | 6 4 3 0 517 | 7 3 3 0 518 | 8 2 3 0 519 | 9 1 3 0 520 |10 0 3 0 521 | 0 11 2 0 522 | 1 10 2 0 523 | 2 9 2 0 524 | 3 8 2 0 525 | 4 7 2 0 526 | 5 6 2 0 527 | 6 5 2 0 528 | 7 4 2 0 529 | 8 3 2 0 530 | 9 2 2 0 531 |10 1 2 0 532 |11 0 2 0 533 | 0 12 1 0 534 | 1 11 1 0 535 | 2 10 1 0 536 | 3 9 1 0 537 | 4 8 1 0 538 | 5 7 1 0 539 | 6 6 1 0 540 | 7 5 1 0 541 | 8 4 1 0 542 | 9 3 1 0 543 |10 2 1 0 544 |11 1 1 0 545 |12 0 1 0 546 | 0 13 0 0 547 | 1 12 0 0 548 | 2 11 0 0 549 | 3 10 0 0 550 | 4 9 0 0 551 | 5 8 0 0 552 | 6 7 0 0 553 | 7 6 0 0 554 | 8 5 0 0 555 | 9 4 0 0 556 |10 3 0 0 557 |11 2 0 0 558 |12 1 0 0 559 |13 0 0 0
 Thomas Andrews (deal@thomasoandrews.com) Copyright 1996-2010. Deal is covered by the GNU General Public License. Plane Dealing graphic above created using POV-Ray.