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of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are assumed to be implicitly extended indefinitely by terms 0.
65:
of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer
369:
1325:
536:
768:
1352:
1185:
1115:
352:
Conventionally the empty composition is counted as the sole composition of 0, and there are no compositions of negative integers. There are 2 compositions of
962:
111:. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the
61:. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same
1461:
270:
It is possible to put constraints on the parts of the compositions. For example the five compositions of 5 into distinct terms are:
548:
1367:
649:
1434:
28:
1214:
1500:
465: − 1 binary choices, the result follows. The same argument shows that the number of compositions of
443:{\displaystyle {\big (}\,\overbrace {1\,\square \,1\,\square \,\ldots \,\square \,1\,\square \,1} ^{n}\,{\big )}}
538:. Note that by summing over all possible numbers of parts we recover 2 as the total number of compositions of
1396:
1278:
1405:
1188:
1354:
are allowed to be zero, then the number of such monomials is exactly the number of weak compositions of
918:{\displaystyle a_{1}+a_{2}+\ldots +a_{k}=n+k\quad \mapsto \quad (a_{1}-1)+(a_{2}-1)+\ldots +(a_{k}-1)=n}
84:
486:
481:
322:
108:
1410:
333:
309:
133:
of the (nonnegative or positive) integers, is an ordered collection of one or more elements in
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62:
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58:
1330:
1128:
1093:
328:
76:
17:
1471:
1391:
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1079:{\displaystyle {\binom {k}{n}}_{(1)_{a\in A}}=\left(\sum _{a\in A}x^{a}\right)^{k}}
1087:
149:
35:
170:
1456:. Discrete Mathematics and its Applications. Boca Raton, Florida: CRC Press.
1427:"Restricted weighted integer compositions and extended binomial coefficients"
80:
54:
1485:
103:, but allowing terms of the sequence to be zero: it is a way of writing
1211:
parts. In fact, a basis for the space is given by the set of monomials
43:
928:
It follows from this formula that the number of weak compositions of
959:
parts is given by the extended binomial (or polynomial) coefficient
327:
308:
169:
148:
75:
290:
Compare this with the three partitions of 5 into distinct terms:
461:
determines an assignment of pluses and commas. Since there are
1086:, where the square brackets indicate the extraction of the
636:{\displaystyle \sum _{k=1}^{n}{n-1 \choose k-1}=2^{n-1}.}
951:-restricted compositions, the number of compositions of
344:
the number of ways one can ascend a staircase of length
1333:
1281:
1217:
1131:
1096:
965:
771:
739:{\displaystyle {n+k-1 \choose k-1}={n+k-1 \choose n}}
652:
551:
489:
372:
359:
Placing either a plus sign or a comma in each of the
1346:
1319:
1267:
1179:
1109:
1078:
917:
738:
635:
530:
442:
1394:(2004). "Compositions of n with parts in a set".
1268:{\displaystyle x_{1}^{d_{1}}\cdots x_{n}^{d_{n}}}
983:
970:
730:
703:
691:
656:
605:
576:
522:
493:
936:parts equals the number of weak compositions of
336:to count the {1, 2}-restricted compositions of
435:
375:
244:Compare this with the seven partitions of 5:
8:
1452:Heubach, Silvia; Mansour, Toufik (2009).
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371:
363: − 1 boxes of the array
1454:Combinatorics of Compositions and Words
1379:
356: ≥ 1; here is a proof:
1203:is the number of weak compositions of
1320:{\displaystyle d_{1}+\ldots +d_{n}=d}
646:For weak compositions, the number is
27:For other uses of "Composition", see
7:
1486:Partition and composition calculator
1117:in the polynomial that follows it.
457:. Conversely, every composition of
348:, taking one or two steps at a time
191:The sixteen compositions of 5 are:
1125:The dimension of the vector space
974:
758:corresponds to a weak one of
707:
660:
580:
497:
321: +1 ordered partitions form
25:
531:{\displaystyle {n-1 \choose k-1}}
453:produces a unique composition of
830:
826:
313:The numbers of compositions of
99:is similar to a composition of
1368:Stars and bars (combinatorics)
1168:
1135:
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997:
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1:
1435:Journal of Integer Sequences
107:as the sum of a sequence of
57:of a sequence of (strictly)
29:Composition (disambiguation)
18:Composition (number theory)
1517:
118:To further generalize, an
26:
1199:variables over the field
153:The 32 compositions of 6
1121:Homogeneous polynomials
123:-restricted composition
73:distinct compositions.
1425:Eger, Steffen (2013).
1397:Congressus Numerantium
1348:
1327:. Since the exponents
1321:
1269:
1189:homogeneous polynomial
1181:
1111:
1080:
919:
740:
637:
572:
532:
444:
349:
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305:Number of compositions
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174:The 11 partitions of 6
167:
88:
1349:
1347:{\displaystyle d_{i}}
1322:
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1182:
1180:{\displaystyle K_{d}}
1112:
1110:{\displaystyle x^{n}}
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741:
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331:
312:
177:1 + 1 + 1 + 1 + 1 + 1
173:
156:1 + 1 + 1 + 1 + 1 + 1
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109:non-negative integers
87:and compositions of 4
79:
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963:
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482:binomial coefficient
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49:is a way of writing
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1501:Integer partitions
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1317:
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1218:
1177:
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480:) is given by the
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334:Fibonacci sequence
326:
266:1 + 1 + 1 + 1 + 1.
240:1 + 1 + 1 + 1 + 1.
189:
168:
89:
1463:978-1-4200-7267-9
1039:
981:
940:− 1 into exactly
728:
689:
603:
520:
431:
424:
323:Pascal's triangle
179:2 + 1 + 1 + 1 + 1
160:1 + 2 + 1 + 1 + 1
158:2 + 1 + 1 + 1 + 1
63:integer partition
59:positive integers
16:(Redirected from
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93:weak composition
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1411:10.1.1.484.5148
1392:Mansour, Toufik
1388:Heubach, Silvia
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1480:External links
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125:of an integer
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85:binary numbers
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955:into exactly
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469:into exactly
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263:2 + 1 + 1 + 1
262:
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245:
239:
237:1 + 1 + 1 + 2
236:
234:1 + 1 + 2 + 1
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228:1 + 2 + 1 + 1
227:
224:
221:
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216:2 + 1 + 1 + 1
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206:
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181:3 + 1 + 1 + 1
172:
151:
144:
142:
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137:whose sum is
136:
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74:
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45:
41:
37:
30:
19:
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1355:
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1204:
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1196:
1192:
1124:
956:
952:
948:
946:
941:
937:
933:
929:
927:
762:by the rule
759:
755:
751:
747:
645:
539:
478:-composition
475:
474:
470:
466:
462:
458:
454:
452:
360:
358:
353:
351:
345:
342:for example,
337:
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138:
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117:
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96:
92:
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66:
50:
46:
39:
33:
1088:coefficient
944:+ 1 parts.
40:composition
36:mathematics
1472:1184.68373
1374:References
1275:such that
1191:of degree
332:Using the
1406:CiteSeerX
1404:: 33–51.
1296:…
1241:⋯
1152:…
1048:∈
1041:∑
1005:∈
901:−
882:…
870:−
845:−
828:↦
799:…
719:−
683:−
672:−
623:−
597:−
586:−
554:∑
514:−
503:−
473:parts (a
422:⏞
414:◻
406:◻
402:…
398:◻
390:◻
260:2 + 2 + 1
257:3 + 1 + 1
231:1 + 1 + 3
225:1 + 2 + 2
222:1 + 3 + 1
213:2 + 1 + 2
210:2 + 2 + 1
204:3 + 1 + 1
81:Bijection
1495:Category
1362:See also
145:Examples
53:as the
44:integer
1470:
1460:
1408:
300:3 + 2.
286:1 + 4.
42:of an
1430:(PDF)
1207:into
297:4 + 1
283:2 + 3
280:3 + 2
277:4 + 1
254:3 + 2
251:4 + 1
219:1 + 4
207:2 + 3
201:3 + 2
198:4 + 1
185:3 + 3
183:. . .
164:1 + 5
162:. . .
1458:ISBN
947:For
69:has
38:, a
1468:Zbl
1402:168
1195:in
1187:of
1090:of
113:end
55:sum
34:In
1497::
1466:.
1440:16
1438:.
1432:.
1400:.
1390:;
1358:.
542::
340:,
141:.
91:A
1474:.
1442:.
1414:.
1356:d
1340:i
1336:d
1315:d
1312:=
1307:n
1303:d
1299:+
1293:+
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1259:n
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1245:x
1234:1
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1209:n
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1197:n
1193:d
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1136:[
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1018:[
1015:=
1008:A
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979:n
976:k
971:(
957:k
953:n
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938:k
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913:n
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904:1
896:k
892:a
888:(
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879:+
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865:2
861:a
857:(
854:+
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848:1
840:1
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832:(
824:k
821:+
818:n
815:=
810:k
806:a
802:+
796:+
791:2
787:a
783:+
778:1
774:a
760:n
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752:n
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726:n
722:1
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710:n
704:(
698:=
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686:1
680:k
675:1
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666:+
663:n
657:(
631:.
626:1
620:n
616:2
612:=
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600:1
594:k
589:1
583:n
577:(
569:n
564:1
561:=
558:k
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517:1
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506:1
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494:(
476:k
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463:n
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376:(
361:n
354:n
346:n
338:n
319:k
315:n
294:5
274:5
248:5
195:5
187:6
166:6
139:n
135:A
131:A
127:n
121:A
105:n
101:n
97:n
71:2
67:n
51:n
47:n
31:.
20:)
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