194:
Bender–Knuth involutions can be used to show that the number of semistandard skew tableaux of given shape and weight is unchanged under permutations of the weight. In turn this implies that the
58:, and act on the set of semistandard skew Young tableaux of some fixed shape μ/ν, where μ and ν are partitions. It acts by changing some of the elements
321:
316:
206:
311:
25:
17:
326:
29:
283:
240:
275:
232:
195:
295:
252:
291:
260:
248:
37:
305:
236:
220:
287:
244:
279:
82: + 1 are exchanged. Call an entry of the tableau
74:, in such a way that the numbers of elements with values
94: + 1 and there is no other element with value
261:"A concise proof of the Littlewood–Richardson rule"
110:are all in consecutive columns, and consist of
8:
223:(1972), "Enumeration of plane partitions",
102: + 1 in the same column. For any
33:
202:
225:Journal of Combinatorial Theory, Series A
198:of a partition is a symmetric function.
66: + 1, and some of the entries
201:Bender–Knuth involutions were used by
7:
36:, pp. 46–47) in their study of
268:Electronic Journal of Combinatorics
14:
274:(1): Note 5, 4 pp. (electronic),
154:. The Bender–Knuth involution σ
48:The Bender–Knuth involutions σ
1:
205:to give a short proof of the
259:Stembridge, John R. (2002),
237:10.1016/0097-3165(72)90007-6
343:
207:Littlewood–Richardson rule
106:, the free entries of row
136: + 1, for some
54:are defined for integers
322:Combinatorial algorithms
34:Bender & Knuth (1972
317:Algebraic combinatorics
22:Bender–Knuth involution
18:algebraic combinatorics
30:semistandard tableaux
312:Symmetric functions
219:Bender, Edward A.;
160:replaces them by
70: + 1 to
62:of the tableau to
203:Stembridge (2002)
334:
298:
265:
255:
221:Knuth, Donald E.
186: + 1.
38:plane partitions
32:, introduced by
342:
341:
337:
336:
335:
333:
332:
331:
302:
301:
263:
258:
218:
215:
192:
181:
168:
159:
153:
144:
131:
118:
53:
46:
12:
11:
5:
340:
338:
330:
329:
324:
319:
314:
304:
303:
300:
299:
256:
214:
211:
196:Schur function
191:
188:
177:
164:
155:
149:
140:
127:
114:
49:
45:
42:
28:on the set of
13:
10:
9:
6:
4:
3:
2:
339:
328:
325:
323:
320:
318:
315:
313:
310:
309:
307:
297:
293:
289:
285:
281:
280:10.37236/1666
277:
273:
269:
262:
257:
254:
250:
246:
242:
238:
234:
230:
226:
222:
217:
216:
212:
210:
208:
204:
199:
197:
189:
187:
185:
180:
176:
172:
167:
163:
158:
152:
148:
143:
139:
135:
130:
126:
122:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
52:
43:
41:
39:
35:
31:
27:
23:
19:
327:Permutations
271:
267:
231:(1): 40–54,
228:
224:
200:
193:
190:Applications
183:
178:
174:
173:followed by
170:
165:
161:
156:
150:
146:
141:
137:
133:
128:
124:
123:followed by
120:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
50:
47:
21:
15:
306:Categories
213:References
182:copies of
169:copies of
132:copies of
119:copies of
44:Definition
26:involution
288:1077-8926
245:1096-0899
86:if it is
296:1912814
253:0299574
294:
286:
251:
243:
24:is an
264:(PDF)
284:ISSN
241:ISSN
145:and
84:free
20:, a
276:doi
233:doi
98:or
90:or
78:or
16:In
308::
292:MR
290:,
282:,
270:,
266:,
249:MR
247:,
239:,
229:13
227:,
209:.
40:.
278::
272:9
235::
184:k
179:i
175:a
171:k
166:i
162:b
157:k
151:i
147:b
142:i
138:a
134:k
129:i
125:b
121:k
116:i
112:a
108:i
104:i
100:k
96:k
92:k
88:k
80:k
76:k
72:k
68:k
64:k
60:k
56:k
51:k
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.