id	sid	eid	entity	type
ejpam-5240	1	1	european journal	ORG
ejpam-5240	2	1	17	CARDINAL
ejpam-5240	2	2	3	CARDINAL
ejpam-5240	2	3	2024	DATE
ejpam-5240	2	4	1779-1803	DATE
ejpam-5240	2	5	1307-5543	DATE
ejpam-5240	2	6	new york	GPE
ejpam-5240	2	7	eliz d. pelagio1∗	PERSON
ejpam-5240	2	8	kathlen c. mendoza1	PERSON
ejpam-5240	2	9	neil m. mame1 1	PERSON
ejpam-5240	2	10	batangas state university	ORG
ejpam-5240	2	11	the national engineering university	ORG
ejpam-5240	2	12	batangas city	GPE
ejpam-5240	2	13	batangas	GPE
ejpam-5240	2	14	philippines	GPE
ejpam-5240	3	1	1945	DATE
ejpam-5240	5	1	sn	ORG
ejpam-5240	5	2	sn	ORG
ejpam-5240	5	3	0 ≤ k ≤ n.	QUANTITY
ejpam-5240	5	4	two	CARDINAL
ejpam-5240	8	1	2020	DATE
ejpam-5240	8	2	05c62	CARDINAL
ejpam-5240	8	3	05c69	CARDINAL
ejpam-5240	8	4	1	CARDINAL
ejpam-5240	10	1	m.	PERSON
ejpam-5240	10	2	1980	DATE
ejpam-5240	10	3	5	CARDINAL
ejpam-5240	11	1	richard hamming	PERSON
ejpam-5240	11	2	johnson	PERSON
ejpam-5240	11	3	selmer m. john	PERSON
ejpam-5240	12	1	1	CARDINAL
ejpam-5240	15	1	two	CARDINAL
ejpam-5240	16	1	marianeeliz.pelagio@g.batstate-u.edu.ph	PERSON
ejpam-5240	16	2	m.e. pelagio	ORG
ejpam-5240	16	3	kathlen.mendoza@g.batstate-u.edu.ph	ORG
ejpam-5240	16	4	k. mendoza	PERSON
ejpam-5240	16	5	neil.mame@g.batstate-u.edu.ph	ORG
ejpam-5240	16	6	n. mame	PERSON
ejpam-5240	16	7	1779	DATE
ejpam-5240	17	1	2024	CARDINAL
ejpam-5240	18	1	m.e.	GPE
ejpam-5240	18	2	pelagio	GPE
ejpam-5240	18	3	n. mame	PERSON
ejpam-5240	18	4	k. mendoza	PERSON
ejpam-5240	19	1	j. pure	PERSON
ejpam-5240	19	2	17	CARDINAL
ejpam-5240	19	3	3	CARDINAL
ejpam-5240	19	4	2024	CARDINAL
ejpam-5240	19	5	1779-1803 1780	DATE
ejpam-5240	20	1	goodman	ORG
ejpam-5240	20	2	1966	DATE
ejpam-5240	22	1	11	CARDINAL
ejpam-5240	26	1	k	PERSON
ejpam-5240	26	2	k ≤ n. hence	ORG
ejpam-5240	27	1	two	CARDINAL
ejpam-5240	28	1	first	ORDINAL
ejpam-5240	28	2	second	ORDINAL
ejpam-5240	30	1	chapter 3.	LAW
ejpam-5240	33	1	2	CARDINAL
ejpam-5240	35	1	2.1	CARDINAL
ejpam-5240	37	1	3	CARDINAL
ejpam-5240	37	2	4	CARDINAL
ejpam-5240	37	3	8	CARDINAL
ejpam-5240	39	1	m.e.	GPE
ejpam-5240	39	2	pelagio	GPE
ejpam-5240	39	3	n. mame	PERSON
ejpam-5240	39	4	k. mendoza	PERSON
ejpam-5240	40	1	j. pure	PERSON
ejpam-5240	40	2	17	CARDINAL
ejpam-5240	40	3	3	CARDINAL
ejpam-5240	40	4	2024	CARDINAL
ejpam-5240	40	5	1779-1803 1781	DATE
ejpam-5240	41	1	sn	ORG
ejpam-5240	42	1	sn	ORG
ejpam-5240	43	1	1	CARDINAL
ejpam-5240	45	1	⊆	CARDINAL
ejpam-5240	45	2	∈	ORG
ejpam-5240	48	1	two	CARDINAL
ejpam-5240	50	1	1	CARDINAL
ejpam-5240	52	1	− |t	PERSON
ejpam-5240	55	1	two	CARDINAL
ejpam-5240	57	1	0 ≤ k ≤ n	QUANTITY
ejpam-5240	61	1	sn	ORG
ejpam-5240	61	2	2	CARDINAL
ejpam-5240	62	1	1	CARDINAL
ejpam-5240	62	2	2	CARDINAL
ejpam-5240	63	1	0	CARDINAL
ejpam-5240	64	1	0	CARDINAL
ejpam-5240	65	1	3	CARDINAL
ejpam-5240	66	1	3	CARDINAL
ejpam-5240	66	2	sn	ORG
ejpam-5240	66	3	sn	ORG
ejpam-5240	67	1	sn	ORG
ejpam-5240	68	1	1	CARDINAL
ejpam-5240	69	1	1	CARDINAL
ejpam-5240	69	2	k = n. theorem 1	PERSON
ejpam-5240	70	1	2	CARDINAL
ejpam-5240	70	2	1	CARDINAL
ejpam-5240	70	3	m.e	GPE
ejpam-5240	70	4	pelagio	GPE
ejpam-5240	70	5	n. mame	PERSON
ejpam-5240	70	6	k. mendoza	PERSON
ejpam-5240	71	1	j. pure	PERSON
ejpam-5240	71	2	17	CARDINAL
ejpam-5240	71	3	3	CARDINAL
ejpam-5240	71	4	2024	CARDINAL
ejpam-5240	71	5	1779-1803 1782	DATE
ejpam-5240	71	6	2	CARDINAL
ejpam-5240	72	1	sn	ORG
ejpam-5240	72	2	k ≤ n.	ORG
ejpam-5240	72	3	sn	ORG
ejpam-5240	72	4	sn	ORG
ejpam-5240	73	1	multiset	ORG
ejpam-5240	73	2	s(n	ORG
ejpam-5240	73	3	sn	ORG
ejpam-5240	74	1	k	PERSON
ejpam-5240	75	1	sn	ORG
ejpam-5240	75	2	0	CARDINAL
ejpam-5240	76	1	4	CARDINAL
ejpam-5240	76	2	4	CARDINAL
ejpam-5240	79	1	2.2	CARDINAL
ejpam-5240	81	1	9	CARDINAL
ejpam-5240	82	1	6	CARDINAL
ejpam-5240	82	2	12	CARDINAL
ejpam-5240	82	3	13	CARDINAL
ejpam-5240	90	1	only one	CARDINAL
ejpam-5240	93	1	∈	ORG
ejpam-5240	93	2	x.	GPE
ejpam-5240	101	1	2	CARDINAL
ejpam-5240	101	2	2	CARDINAL
ejpam-5240	101	3	2	CARDINAL
ejpam-5240	101	4	1	CARDINAL
ejpam-5240	101	5	m.e	GPE
ejpam-5240	101	6	pelagio	GPE
ejpam-5240	101	7	n. mame	PERSON
ejpam-5240	101	8	k. mendoza	PERSON
ejpam-5240	102	1	j. pure	PERSON
ejpam-5240	102	2	17	CARDINAL
ejpam-5240	102	3	3	CARDINAL
ejpam-5240	102	4	2024	CARDINAL
ejpam-5240	102	5	1779-1803 1783	DATE
ejpam-5240	104	1	0	CARDINAL
ejpam-5240	109	1	xn−1	GPE
ejpam-5240	110	1	kn	PERSON
ejpam-5240	111	1	kn	PERSON
ejpam-5240	111	2	n − 1	ORG
ejpam-5240	112	1	kn	PERSON
ejpam-5240	113	1	kn	ORG
ejpam-5240	113	2	2	CARDINAL
ejpam-5240	113	3	1	CARDINAL
ejpam-5240	113	4	2	CARDINAL
ejpam-5240	115	1	c3	PERSON
ejpam-5240	115	2	3	CARDINAL
ejpam-5240	115	3	2	CARDINAL
ejpam-5240	116	1	3	CARDINAL
ejpam-5240	116	2	2	CARDINAL
ejpam-5240	116	3	3	CARDINAL
ejpam-5240	118	1	two	CARDINAL
ejpam-5240	119	1	5	CARDINAL
ejpam-5240	120	1	c3	PERSON
ejpam-5240	120	2	3	CARDINAL
ejpam-5240	122	1	3	CARDINAL
ejpam-5240	125	1	1 ≤	QUANTITY
ejpam-5240	129	1	1	CARDINAL
ejpam-5240	129	2	1	CARDINAL
ejpam-5240	129	3	2	DATE
ejpam-5240	129	4	3	CARDINAL
ejpam-5240	129	5	2	CARDINAL
ejpam-5240	129	6	4	DATE
ejpam-5240	129	7	1	CARDINAL
ejpam-5240	129	8	3	CARDINAL
ejpam-5240	130	1	3	CARDINAL
ejpam-5240	131	1	1	CARDINAL
ejpam-5240	131	2	2	DATE
ejpam-5240	131	3	3	CARDINAL
ejpam-5240	131	4	2	CARDINAL
ejpam-5240	131	5	4	CARDINAL
ejpam-5240	131	6	2	CARDINAL
ejpam-5240	131	7	1	CARDINAL
ejpam-5240	131	8	2	DATE
ejpam-5240	131	9	3}∩{1	CARDINAL
ejpam-5240	131	10	3	CARDINAL
ejpam-5240	131	11	1	CARDINAL
ejpam-5240	131	12	3	CARDINAL
ejpam-5240	131	13	1	CARDINAL
ejpam-5240	131	14	2	DATE
ejpam-5240	131	15	3	CARDINAL
ejpam-5240	131	16	2	CARDINAL
ejpam-5240	131	17	4	CARDINAL
ejpam-5240	131	18	1	CARDINAL
ejpam-5240	131	19	2	DATE
ejpam-5240	131	20	3	CARDINAL
ejpam-5240	131	21	1	CARDINAL
ejpam-5240	131	22	3	CARDINAL
ejpam-5240	131	23	2	CARDINAL
ejpam-5240	131	24	4	DATE
ejpam-5240	131	25	1	CARDINAL
ejpam-5240	131	26	3	CARDINAL
ejpam-5240	131	27	2	CARDINAL
ejpam-5240	131	28	4	DATE
ejpam-5240	131	29	1	CARDINAL
ejpam-5240	131	30	3	CARDINAL
ejpam-5240	134	1	1	CARDINAL
ejpam-5240	134	2	2	DATE
ejpam-5240	134	3	3	CARDINAL
ejpam-5240	134	4	2	CARDINAL
ejpam-5240	134	5	4	CARDINAL
ejpam-5240	134	6	1	CARDINAL
ejpam-5240	134	7	2	DATE
ejpam-5240	134	8	3	CARDINAL
ejpam-5240	134	9	1	CARDINAL
ejpam-5240	134	10	3	CARDINAL
ejpam-5240	135	1	1	CARDINAL
ejpam-5240	135	2	1	CARDINAL
ejpam-5240	135	3	2	DATE
ejpam-5240	135	4	3	CARDINAL
ejpam-5240	135	5	2	CARDINAL
ejpam-5240	135	6	4	DATE
ejpam-5240	135	7	1	CARDINAL
ejpam-5240	135	8	3	CARDINAL
ejpam-5240	136	1	m.e	PERSON
ejpam-5240	136	2	pelagio	GPE
ejpam-5240	136	3	n. mame	PERSON
ejpam-5240	136	4	k. mendoza	PERSON
ejpam-5240	137	1	j. pure	PERSON
ejpam-5240	137	2	17	CARDINAL
ejpam-5240	137	3	3	CARDINAL
ejpam-5240	137	4	2024	CARDINAL
ejpam-5240	137	5	1779-1803 1784	DATE
ejpam-5240	137	6	2	CARDINAL
ejpam-5240	137	7	4	DATE
ejpam-5240	137	8	1	CARDINAL
ejpam-5240	137	9	2	CARDINAL
ejpam-5240	137	10	3	CARDINAL
ejpam-5240	137	11	1	CARDINAL
ejpam-5240	142	1	4	CARDINAL
ejpam-5240	144	1	nonempty	ORG
ejpam-5240	144	2	⊆	CARDINAL
ejpam-5240	145	1	α(g	PERSON
ejpam-5240	145	2	⊆	CARDINAL
ejpam-5240	145	3	α(g	PERSON
ejpam-5240	146	1	5	CARDINAL
ejpam-5240	149	1	nonempty	ORG
ejpam-5240	150	1	γ(g	NORP
ejpam-5240	151	1	⊆	CARDINAL
ejpam-5240	151	2	⊆	CARDINAL
ejpam-5240	153	1	6	CARDINAL
ejpam-5240	154	1	at least one	CARDINAL
ejpam-5240	157	1	⊆	CARDINAL
ejpam-5240	157	2	γ(g	GPE
ejpam-5240	159	1	3	CARDINAL
ejpam-5240	160	1	s(n	ORG
ejpam-5240	160	2	sn	ORG
ejpam-5240	161	1	3	CARDINAL
ejpam-5240	163	1	0	CARDINAL
ejpam-5240	163	2	s0	ORG
ejpam-5240	163	3	m.e	GPE
ejpam-5240	163	4	pelagio	GPE
ejpam-5240	163	5	n. mame	PERSON
ejpam-5240	163	6	k. mendoza	PERSON
ejpam-5240	164	1	j. pure	PERSON
ejpam-5240	164	2	17	CARDINAL
ejpam-5240	164	3	3	CARDINAL
ejpam-5240	164	4	2024	CARDINAL
ejpam-5240	164	5	1779-1803	DATE
ejpam-5240	164	6	1785	DATE
ejpam-5240	164	7	sn	ORG
ejpam-5240	164	8	k ≤ n. definition	ORG
ejpam-5240	164	9	7	CARDINAL
ejpam-5240	165	1	sn	ORG
ejpam-5240	165	2	k ≤ n	FAC
ejpam-5240	165	3	sn	ORG
ejpam-5240	166	1	s(n	ORG
ejpam-5240	166	2	two	CARDINAL
ejpam-5240	167	1	sn	ORG
ejpam-5240	167	2	∈	ORG
ejpam-5240	170	1	∈	ORG
ejpam-5240	171	1	2	CARDINAL
ejpam-5240	171	2	4	CARDINAL
ejpam-5240	171	3	2	CARDINAL
ejpam-5240	171	4	2	CARDINAL
ejpam-5240	171	5	x3	GPE
ejpam-5240	171	6	2	CARDINAL
ejpam-5240	172	1	x3}	GPE
ejpam-5240	173	1	x4	PERSON
ejpam-5240	173	2	x3}	GPE
ejpam-5240	179	1	x3, x4}]	ORG
ejpam-5240	179	2	x3, x4}]	ORG
ejpam-5240	179	3	x3}	GPE
ejpam-5240	179	4	x3}	GPE
ejpam-5240	179	5	x3, x4}]	ORG
ejpam-5240	179	6	x3, x4}	ORG
ejpam-5240	180	1	6	CARDINAL
ejpam-5240	180	2	12	CARDINAL
ejpam-5240	181	1	2	CARDINAL
ejpam-5240	183	1	sn	ORG
ejpam-5240	185	1	6	CARDINAL
ejpam-5240	187	1	2	CARDINAL
ejpam-5240	189	1	2	CARDINAL
ejpam-5240	199	1	k. m.e	PERSON
ejpam-5240	199	2	pelagio	GPE
ejpam-5240	199	3	n. mame	PERSON
ejpam-5240	199	4	k. mendoza	PERSON
ejpam-5240	200	1	j. pure	PERSON
ejpam-5240	200	2	17	CARDINAL
ejpam-5240	200	3	3	CARDINAL
ejpam-5240	200	4	2024	CARDINAL
ejpam-5240	200	5	1779-1803 1786	DATE
ejpam-5240	200	6	x3, x4	ORG
ejpam-5240	200	7	x4	PERSON
ejpam-5240	200	8	x4	PERSON
ejpam-5240	200	9	2	CARDINAL
ejpam-5240	200	10	2	CARDINAL
ejpam-5240	201	1	1	CARDINAL
ejpam-5240	202	1	2	CARDINAL
ejpam-5240	202	2	4	CARDINAL
ejpam-5240	202	3	x3	GPE
ejpam-5240	202	4	2	CARDINAL
ejpam-5240	204	1	6	CARDINAL
ejpam-5240	204	2	2	CARDINAL
ejpam-5240	204	3	2	CARDINAL
ejpam-5240	205	1	4 2	CARDINAL
ejpam-5240	206	1	6	CARDINAL
ejpam-5240	208	1	only one	CARDINAL
ejpam-5240	209	1	1	CARDINAL
ejpam-5240	210	1	1	CARDINAL
ejpam-5240	210	2	k = n. theorem	PERSON
ejpam-5240	210	3	3	CARDINAL
ejpam-5240	210	4	0	DATE
ejpam-5240	210	5	3	CARDINAL
ejpam-5240	211	1	sn	ORG
ejpam-5240	216	1	1	CARDINAL
ejpam-5240	218	1	1	CARDINAL
ejpam-5240	218	2	theorem 1	DATE
ejpam-5240	218	3	k = n.	PERSON
ejpam-5240	219	1	gs(n,0	ORG
ejpam-5240	222	1	gs(n,0	ORG
ejpam-5240	223	1	k = n	PERSON
ejpam-5240	223	2	s(n	ORG
ejpam-5240	223	3	sn	ORG
ejpam-5240	224	1	sn	ORG
ejpam-5240	225	1	only one	CARDINAL
ejpam-5240	226	1	6	CARDINAL
ejpam-5240	226	2	sn	ORG
ejpam-5240	226	3	sn	ORG
ejpam-5240	229	1	2	CARDINAL
ejpam-5240	230	1	x3	GPE
ejpam-5240	231	1	gs(4,0	PERSON
ejpam-5240	233	1	4	CARDINAL
ejpam-5240	234	1	x3	GPE
ejpam-5240	235	1	3	CARDINAL
ejpam-5240	236	1	1	CARDINAL
ejpam-5240	236	2	1	CARDINAL
ejpam-5240	236	3	sn	ORG
ejpam-5240	236	4	4	CARDINAL
ejpam-5240	236	5	1	CARDINAL
ejpam-5240	236	6	4	CARDINAL
ejpam-5240	237	1	sn	ORG
ejpam-5240	238	1	1	CARDINAL
ejpam-5240	238	2	n. m.e	PERSON
ejpam-5240	238	3	pelagio	GPE
ejpam-5240	238	4	n. mame	PERSON
ejpam-5240	238	5	k. mendoza	PERSON
ejpam-5240	239	1	j. pure	PERSON
ejpam-5240	239	2	17	CARDINAL
ejpam-5240	239	3	3	CARDINAL
ejpam-5240	239	4	2024	CARDINAL
ejpam-5240	239	5	1779-1803 1787	DATE
ejpam-5240	239	6	x3	GPE
ejpam-5240	239	7	3	CARDINAL
ejpam-5240	239	8	gs(4,0	PERSON
ejpam-5240	241	1	1	CARDINAL
ejpam-5240	247	1	gs(n,0	ORG
ejpam-5240	248	1	3	CARDINAL
ejpam-5240	249	1	x3	GPE
ejpam-5240	249	2	k = 1.	PERSON
ejpam-5240	252	1	1	CARDINAL
ejpam-5240	253	1	e(gs(4,1	PERSON
ejpam-5240	254	1	4	CARDINAL
ejpam-5240	254	2	4	CARDINAL
ejpam-5240	255	1	4	CARDINAL
ejpam-5240	256	1	nonnegative	ORG
ejpam-5240	256	2	k. lemma 1	PERSON
ejpam-5240	256	3	1 ≤ k ≤	QUANTITY
ejpam-5240	256	4	2	CARDINAL
ejpam-5240	257	1	1	CARDINAL
ejpam-5240	258	1	sn	ORG
ejpam-5240	259	1	∈	ORG
ejpam-5240	259	2	deg(a	PERSON
ejpam-5240	261	1	1 ≤ k ≤	QUANTITY
ejpam-5240	261	2	2	CARDINAL
ejpam-5240	263	1	deg(a	PERSON
ejpam-5240	266	1	k ≤ n. if k	PERSON
ejpam-5240	267	1	n−0 0	DATE
ejpam-5240	268	1	1	CARDINAL
ejpam-5240	268	2	1 + 1	DATE
ejpam-5240	268	3	−1	DATE
ejpam-5240	270	1	k ≤ n	PERSON
ejpam-5240	270	2	2	CARDINAL
ejpam-5240	271	1	n − k ≤ ⌊ n 2	PERSON
ejpam-5240	271	2	4	CARDINAL
ejpam-5240	273	1	deg(a	PERSON
ejpam-5240	275	1	0 + 1	DATE
ejpam-5240	277	1	1	CARDINAL
ejpam-5240	278	1	deg(a	PERSON
ejpam-5240	282	1	1 ≤ k ≤	QUANTITY
ejpam-5240	282	2	2	CARDINAL
ejpam-5240	283	1	1 ≤ k ≤	QUANTITY
ejpam-5240	284	1	⊆	CARDINAL
ejpam-5240	284	2	sn	ORG
ejpam-5240	284	3	1	CARDINAL
ejpam-5240	284	4	− k.	PERSON
ejpam-5240	284	5	sn \ a.	ORG
ejpam-5240	284	6	⊆	CARDINAL
ejpam-5240	285	1	1 ≤ k ≤	QUANTITY
ejpam-5240	287	1	k ≤ n− k so |s(n−k	ORG
ejpam-5240	290	1	2	CARDINAL
ejpam-5240	290	2	deg(a	PERSON
ejpam-5240	293	1	6	CARDINAL
ejpam-5240	295	1	m.e	GPE
ejpam-5240	295	2	pelagio	GPE
ejpam-5240	295	3	n. mame	PERSON
ejpam-5240	295	4	k. mendoza	PERSON
ejpam-5240	296	1	j. pure	PERSON
ejpam-5240	296	2	17	CARDINAL
ejpam-5240	296	3	3	CARDINAL
ejpam-5240	296	4	2024	CARDINAL
ejpam-5240	296	5	1779-1803 1788	DATE
ejpam-5240	296	6	deg(a	PERSON
ejpam-5240	300	1	1 ≤ k ≤	QUANTITY
ejpam-5240	303	1	4	CARDINAL
ejpam-5240	304	1	x3	GPE
ejpam-5240	304	2	x4	GPE
ejpam-5240	304	3	2	CARDINAL
ejpam-5240	306	1	x5}	GPE
ejpam-5240	306	2	x3}	GPE
ejpam-5240	306	3	x5}	GPE
ejpam-5240	306	4	x3	GPE
ejpam-5240	306	5	x5}	GPE
ejpam-5240	306	6	x4	PERSON
ejpam-5240	306	7	5	CARDINAL
ejpam-5240	307	1	x3, x5	ORG
ejpam-5240	307	2	x4	PERSON
ejpam-5240	307	3	x3, x4	ORG
ejpam-5240	307	4	x4	PERSON
ejpam-5240	307	5	x4	PERSON
ejpam-5240	307	6	5	CARDINAL
ejpam-5240	308	1	6	CARDINAL
ejpam-5240	309	1	2	CARDINAL
ejpam-5240	309	2	k ≤	FAC
ejpam-5240	309	3	5 2	CARDINAL
ejpam-5240	309	4	2	CARDINAL
ejpam-5240	309	5	1	CARDINAL
ejpam-5240	309	6	2	CARDINAL
ejpam-5240	309	7	∈	ORG
ejpam-5240	309	8	deg(a	PERSON
ejpam-5240	310	1	k k	PERSON
ejpam-5240	311	1	5 2	CARDINAL
ejpam-5240	312	1	5− 2 2	DATE
ejpam-5240	313	1	5 2	CARDINAL
ejpam-5240	314	1	3 2	CARDINAL
ejpam-5240	314	2	10−	CARDINAL
ejpam-5240	314	3	3 + 1	DATE
ejpam-5240	314	4	6	CARDINAL
ejpam-5240	314	5	m.e	GPE
ejpam-5240	314	6	pelagio	GPE
ejpam-5240	314	7	n. mame	PERSON
ejpam-5240	314	8	k. mendoza	PERSON
ejpam-5240	315	1	j. pure	PERSON
ejpam-5240	315	2	17	CARDINAL
ejpam-5240	315	3	3	CARDINAL
ejpam-5240	315	4	2024	CARDINAL
ejpam-5240	315	5	1779-1803 1789	DATE
ejpam-5240	315	6	2	CARDINAL
ejpam-5240	315	7	0	CARDINAL
ejpam-5240	315	8	k ≤ n. lemma	PERSON
ejpam-5240	316	1	2	CARDINAL
ejpam-5240	316	2	sn	ORG
ejpam-5240	317	1	deg(a	PERSON
ejpam-5240	318	1	∈	ORG
ejpam-5240	319	1	0	CARDINAL
ejpam-5240	320	1	deg(a	PERSON
ejpam-5240	321	1	−1	DATE
ejpam-5240	322	1	1 < k	QUANTITY
ejpam-5240	323	1	lemma 1	ORG
ejpam-5240	323	2	deg(a	PERSON
ejpam-5240	328	1	1	CARDINAL
ejpam-5240	329	1	0	CARDINAL
ejpam-5240	332	1	1 < k	QUANTITY
ejpam-5240	336	1	0	CARDINAL
ejpam-5240	336	2	k ≤ n. conversely	PERSON
ejpam-5240	336	3	k ≤ n. if k	PERSON
ejpam-5240	336	4	0	CARDINAL
ejpam-5240	336	5	3	CARDINAL
ejpam-5240	338	1	1	CARDINAL
ejpam-5240	338	2	0	DATE
ejpam-5240	338	3	deg(a	PERSON
ejpam-5240	339	1	1	CARDINAL
ejpam-5240	339	2	1	CARDINAL
ejpam-5240	340	1	1	CARDINAL
ejpam-5240	340	2	∈	ORG
ejpam-5240	342	1	⊆	CARDINAL
ejpam-5240	342	2	sn	ORG
ejpam-5240	342	3	1	CARDINAL
ejpam-5240	342	4	\a|	DATE
ejpam-5240	343	1	sn	ORG
ejpam-5240	344	1	⊆	CARDINAL
ejpam-5240	345	1	4, |s(n−k	PERCENT
ejpam-5240	345	2	n−k k	PERSON
ejpam-5240	346	1	0	CARDINAL
ejpam-5240	348	1	deg(a	PERSON
ejpam-5240	349	1	0	CARDINAL
ejpam-5240	350	1	deg(a	PERSON
ejpam-5240	353	1	1	CARDINAL
ejpam-5240	355	1	1	CARDINAL
ejpam-5240	355	2	5	CARDINAL
ejpam-5240	355	3	0	DATE
ejpam-5240	355	4	4	CARDINAL
ejpam-5240	355	5	3	CARDINAL
ejpam-5240	355	6	5	CARDINAL
ejpam-5240	355	7	x3	GPE
ejpam-5240	356	1	0	CARDINAL
ejpam-5240	356	2	k ≤	PERSON
ejpam-5240	357	1	3	CARDINAL
ejpam-5240	358	1	0	CARDINAL
ejpam-5240	358	2	2	CARDINAL
ejpam-5240	359	1	4	CARDINAL
ejpam-5240	360	1	1	CARDINAL
ejpam-5240	360	2	1− 1	CARDINAL
ejpam-5240	360	3	0	CARDINAL
ejpam-5240	361	1	4	CARDINAL
ejpam-5240	361	2	3	CARDINAL
ejpam-5240	362	1	x3	GPE
ejpam-5240	362	2	3	CARDINAL
ejpam-5240	363	1	x3	GPE
ejpam-5240	364	1	0	CARDINAL
ejpam-5240	364	2	2	CARDINAL
ejpam-5240	364	3	4	CARDINAL
ejpam-5240	364	4	deg({x1	ORG
ejpam-5240	364	5	x3	GPE
ejpam-5240	364	6	4 4	DATE
ejpam-5240	365	1	1	CARDINAL
ejpam-5240	365	2	1− 1	CARDINAL
ejpam-5240	365	3	0	CARDINAL
ejpam-5240	366	1	3	CARDINAL
ejpam-5240	367	1	x3}	GPE
ejpam-5240	367	2	x3	GPE
ejpam-5240	368	1	6	CARDINAL
ejpam-5240	369	1	deg(a	PERSON
ejpam-5240	369	2	3	CARDINAL
ejpam-5240	369	3	∈	ORG
ejpam-5240	370	1	3	CARDINAL
ejpam-5240	370	2	2	CARDINAL
ejpam-5240	370	3	2	CARDINAL
ejpam-5240	370	4	3	CARDINAL
ejpam-5240	370	5	m.e	GPE
ejpam-5240	370	6	pelagio	GPE
ejpam-5240	370	7	n. mame	PERSON
ejpam-5240	370	8	k. mendoza	PERSON
ejpam-5240	371	1	j. pure	PERSON
ejpam-5240	371	2	17	CARDINAL
ejpam-5240	371	3	3	CARDINAL
ejpam-5240	371	4	2024	CARDINAL
ejpam-5240	371	5	1779-1803 1790	DATE
ejpam-5240	371	6	x4	PERSON
ejpam-5240	371	7	x3	GPE
ejpam-5240	371	8	6	CARDINAL
ejpam-5240	372	1	deg(a	PERSON
ejpam-5240	374	1	4 3	DATE
ejpam-5240	374	2	1	CARDINAL
ejpam-5240	374	3	3	CARDINAL
ejpam-5240	375	1	1	CARDINAL
ejpam-5240	375	2	lemma 2	ORG
ejpam-5240	378	1	5	CARDINAL
ejpam-5240	378	2	5	CARDINAL
ejpam-5240	382	1	1 ≤ k ≤	QUANTITY
ejpam-5240	382	2	1	CARDINAL
ejpam-5240	383	1	0	CARDINAL
ejpam-5240	384	1	1	CARDINAL
ejpam-5240	384	2	lemma 2	ORG
ejpam-5240	385	1	1	CARDINAL
ejpam-5240	386	1	1 ≤ k ≤	QUANTITY
ejpam-5240	386	2	2	CARDINAL
ejpam-5240	386	3	1	CARDINAL
ejpam-5240	386	4	deg(a	PERSON
ejpam-5240	388	1	1	CARDINAL
ejpam-5240	388	2	∈	ORG
ejpam-5240	389	1	5	CARDINAL
ejpam-5240	389	2	0	CARDINAL
ejpam-5240	390	1	0	CARDINAL
ejpam-5240	390	2	k = n. equivalently	ORG
ejpam-5240	391	1	7	CARDINAL
ejpam-5240	392	1	0	CARDINAL
ejpam-5240	392	2	0	CARDINAL
ejpam-5240	392	3	1	CARDINAL
ejpam-5240	393	1	1	CARDINAL
ejpam-5240	393	2	2	CARDINAL
ejpam-5240	395	1	1	CARDINAL
ejpam-5240	395	2	6	CARDINAL
ejpam-5240	398	1	2	CARDINAL
ejpam-5240	398	2	1 ≤ k ≤	QUANTITY
ejpam-5240	398	3	2	CARDINAL
ejpam-5240	398	4	k ≤ n. m.e	PERSON
ejpam-5240	398	5	pelagio	GPE
ejpam-5240	398	6	n. mame	PERSON
ejpam-5240	398	7	k. mendoza	PERSON
ejpam-5240	399	1	j. pure	PERSON
ejpam-5240	399	2	17	CARDINAL
ejpam-5240	399	3	3	CARDINAL
ejpam-5240	399	4	2024	CARDINAL
ejpam-5240	399	5	1779-1803	DATE
ejpam-5240	401	1	5	CARDINAL
ejpam-5240	402	1	1	CARDINAL
ejpam-5240	402	2	2	CARDINAL
ejpam-5240	403	1	1 ≤ k ≤	QUANTITY
ejpam-5240	403	2	2	CARDINAL
ejpam-5240	406	1	|e(gs(n	PERSON
ejpam-5240	406	2	2	CARDINAL
ejpam-5240	407	1	0	CARDINAL
ejpam-5240	407	2	k ≤ n	PERSON
ejpam-5240	408	1	1	CARDINAL
ejpam-5240	409	1	|e(gs(n	PERSON
ejpam-5240	411	1	2	CARDINAL
ejpam-5240	411	2	6	CARDINAL
ejpam-5240	411	3	5	CARDINAL
ejpam-5240	411	4	2	CARDINAL
ejpam-5240	411	5	5 2	CARDINAL
ejpam-5240	411	6	2	CARDINAL
ejpam-5240	411	7	10	CARDINAL
ejpam-5240	411	8	6	CARDINAL
ejpam-5240	411	9	6	CARDINAL
ejpam-5240	411	10	|e(gs(5,2	GPE
ejpam-5240	412	1	2	CARDINAL
ejpam-5240	412	2	30	CARDINAL
ejpam-5240	413	1	6	CARDINAL
ejpam-5240	413	2	3	CARDINAL
ejpam-5240	413	3	4 3	DATE
ejpam-5240	413	4	4	CARDINAL
ejpam-5240	413	5	3	CARDINAL
ejpam-5240	413	6	4 2	CARDINAL
ejpam-5240	413	7	2	CARDINAL
ejpam-5240	415	1	4(3	CARDINAL
ejpam-5240	415	2	2	CARDINAL
ejpam-5240	415	3	6	CARDINAL
ejpam-5240	416	1	0	CARDINAL
ejpam-5240	417	1	1	CARDINAL
ejpam-5240	417	2	k = n	PERSON
ejpam-5240	417	3	7	CARDINAL
ejpam-5240	418	1	2	CARDINAL
ejpam-5240	418	2	0	CARDINAL
ejpam-5240	419	1	8	CARDINAL
ejpam-5240	419	2	8	CARDINAL
ejpam-5240	420	1	1	CARDINAL
ejpam-5240	420	2	k = n	PERSON
ejpam-5240	421	1	0	CARDINAL
ejpam-5240	421	2	4	CARDINAL
ejpam-5240	422	1	integer k.	PERSON
ejpam-5240	424	1	7	CARDINAL
ejpam-5240	426	1	0	CARDINAL
ejpam-5240	429	1	1	CARDINAL
ejpam-5240	429	2	5	CARDINAL
ejpam-5240	429	3	0	CARDINAL
ejpam-5240	429	4	k ≤ n. conversely	PERSON
ejpam-5240	429	5	5	CARDINAL
ejpam-5240	430	1	1	CARDINAL
ejpam-5240	433	1	∈	ORG
ejpam-5240	434	1	1	CARDINAL
ejpam-5240	434	2	gs(n,0	ORG
ejpam-5240	434	3	1	CARDINAL
ejpam-5240	434	4	7	CARDINAL
ejpam-5240	435	1	x3	GPE
ejpam-5240	435	2	x4	GPE
ejpam-5240	435	3	3	CARDINAL
ejpam-5240	437	1	x3}	GPE
ejpam-5240	437	2	x5}	GPE
ejpam-5240	437	3	x3	GPE
ejpam-5240	437	4	x5}	GPE
ejpam-5240	437	5	x4	PERSON
ejpam-5240	437	6	x5}	GPE
ejpam-5240	437	7	x3	GPE
ejpam-5240	437	8	x3	GPE
ejpam-5240	437	9	x5}	GPE
ejpam-5240	437	10	x4	PERSON
ejpam-5240	437	11	x5}	GPE
ejpam-5240	437	12	x3, x4	ORG
ejpam-5240	438	1	m.e	PERSON
ejpam-5240	438	2	pelagio	GPE
ejpam-5240	438	3	n. mame	PERSON
ejpam-5240	438	4	k. mendoza	PERSON
ejpam-5240	439	1	j. pure	PERSON
ejpam-5240	439	2	17	CARDINAL
ejpam-5240	439	3	3	CARDINAL
ejpam-5240	439	4	2024	CARDINAL
ejpam-5240	439	5	1779-1803 1792	DATE
ejpam-5240	439	6	7	CARDINAL
ejpam-5240	440	1	x4	PERSON
ejpam-5240	440	2	x3	GPE
ejpam-5240	440	3	x4	PERSON
ejpam-5240	440	4	x4	PERSON
ejpam-5240	440	5	x3	GPE
ejpam-5240	440	6	x3, x4	ORG
ejpam-5240	440	7	x3	GPE
ejpam-5240	440	8	7	CARDINAL
ejpam-5240	442	1	k = 3	PERSON
ejpam-5240	442	2	5 2	CARDINAL
ejpam-5240	442	3	2	CARDINAL
ejpam-5240	442	4	3	CARDINAL
ejpam-5240	442	5	5 3	CARDINAL
ejpam-5240	443	1	10	CARDINAL
ejpam-5240	445	1	8	CARDINAL
ejpam-5240	446	1	3	CARDINAL
ejpam-5240	446	2	2	CARDINAL
ejpam-5240	447	1	3	CARDINAL
ejpam-5240	448	1	2	CARDINAL
ejpam-5240	450	1	3	CARDINAL
ejpam-5240	451	1	3	CARDINAL
ejpam-5240	452	1	1 or	CARDINAL
ejpam-5240	452	2	4	CARDINAL
ejpam-5240	453	1	3	CARDINAL
ejpam-5240	454	1	0	CARDINAL
ejpam-5240	454	2	1	CARDINAL
ejpam-5240	454	3	3	CARDINAL
ejpam-5240	454	4	4	DATE
ejpam-5240	455	1	3	CARDINAL
ejpam-5240	456	1	2	CARDINAL
ejpam-5240	456	2	1	CARDINAL
ejpam-5240	456	3	4	CARDINAL
ejpam-5240	457	1	3	CARDINAL
ejpam-5240	458	1	2	CARDINAL
ejpam-5240	459	1	2	CARDINAL
ejpam-5240	459	2	3 2	CARDINAL
ejpam-5240	459	3	2	CARDINAL
ejpam-5240	459	4	7	CARDINAL
ejpam-5240	459	5	3 2	CARDINAL
ejpam-5240	460	1	3	CARDINAL
ejpam-5240	460	2	5	CARDINAL
ejpam-5240	460	3	3	CARDINAL
ejpam-5240	461	1	3	CARDINAL
ejpam-5240	462	1	2	CARDINAL
ejpam-5240	462	2	pelagio	GPE
ejpam-5240	462	3	n. mame	PERSON
ejpam-5240	462	4	k. mendoza	PERSON
ejpam-5240	463	1	j. pure	PERSON
ejpam-5240	463	2	17	CARDINAL
ejpam-5240	463	3	3	CARDINAL
ejpam-5240	463	4	2024	CARDINAL
ejpam-5240	463	5	1779-1803	DATE
ejpam-5240	463	6	3	CARDINAL
ejpam-5240	463	7	3	CARDINAL
ejpam-5240	464	1	3	CARDINAL
ejpam-5240	465	1	8	CARDINAL
ejpam-5240	466	1	2	CARDINAL
ejpam-5240	468	1	3	CARDINAL
ejpam-5240	469	1	k = 3	ORG
ejpam-5240	469	2	3 2	CARDINAL
ejpam-5240	469	3	3	CARDINAL
ejpam-5240	470	1	5	CARDINAL
ejpam-5240	471	1	3	CARDINAL
ejpam-5240	471	2	8	CARDINAL
ejpam-5240	472	1	8	CARDINAL
ejpam-5240	476	1	9	CARDINAL
ejpam-5240	478	1	0	CARDINAL
ejpam-5240	479	1	k ≤ n	PERSON
ejpam-5240	480	1	1 ≤ k ≤	QUANTITY
ejpam-5240	482	1	5	CARDINAL
ejpam-5240	484	1	1	CARDINAL
ejpam-5240	484	2	0	CARDINAL
ejpam-5240	484	3	k ≤ n	PERSON
ejpam-5240	486	1	deg(a	PERSON
ejpam-5240	487	1	∈	ORG
ejpam-5240	488	1	2	CARDINAL
ejpam-5240	489	1	1 ≤ k ≤	QUANTITY
ejpam-5240	489	2	2	CARDINAL
ejpam-5240	489	3	1	CARDINAL
ejpam-5240	489	4	deg(a	PERSON
ejpam-5240	492	1	∈	ORG
ejpam-5240	495	1	deg(a	PERSON
ejpam-5240	496	1	∈	ORG
ejpam-5240	497	1	9	CARDINAL
ejpam-5240	498	1	m.e.	ORG
ejpam-5240	498	2	pelagio	GPE
ejpam-5240	498	3	n. mame	PERSON
ejpam-5240	498	4	k. mendoza	PERSON
ejpam-5240	499	1	j. pure	PERSON
ejpam-5240	499	2	17	CARDINAL
ejpam-5240	499	3	3	CARDINAL
ejpam-5240	499	4	2024	CARDINAL
ejpam-5240	499	5	1779-1803 1794	DATE
ejpam-5240	500	1	x3	GPE
ejpam-5240	500	2	3	CARDINAL
ejpam-5240	500	3	9	CARDINAL
ejpam-5240	501	1	k = 3	PERSON
ejpam-5240	501	2	4 2	CARDINAL
ejpam-5240	501	3	2	CARDINAL
ejpam-5240	501	4	4 3	DATE
ejpam-5240	502	1	4	CARDINAL
ejpam-5240	503	1	4 3	DATE
ejpam-5240	504	1	4	CARDINAL
ejpam-5240	504	2	9	CARDINAL
ejpam-5240	505	1	x4	PERSON
ejpam-5240	505	2	x3	GPE
ejpam-5240	505	3	x4	PERSON
ejpam-5240	505	4	x3	GPE
ejpam-5240	505	5	9	CARDINAL
ejpam-5240	506	1	deg(a	PERSON
ejpam-5240	506	2	∈	ORG
ejpam-5240	507	1	0	CARDINAL
ejpam-5240	508	1	1 ≤ k ≤	QUANTITY
ejpam-5240	508	2	2	CARDINAL
ejpam-5240	508	3	x3	GPE
ejpam-5240	508	4	2	CARDINAL
ejpam-5240	508	5	2	CARDINAL
ejpam-5240	509	1	10	CARDINAL
ejpam-5240	510	1	x3, x4	ORG
ejpam-5240	510	2	x4	PERSON
ejpam-5240	510	3	x4	PERSON
ejpam-5240	510	4	10	CARDINAL
ejpam-5240	511	1	x3, x4}]	ORG
ejpam-5240	511	2	x3}	GPE
ejpam-5240	512	1	1	CARDINAL
ejpam-5240	513	1	4 2	CARDINAL
ejpam-5240	513	2	2	CARDINAL
ejpam-5240	513	3	9	CARDINAL
ejpam-5240	513	4	2	CARDINAL
ejpam-5240	513	5	pelagio	GPE
ejpam-5240	513	6	n. mame	PERSON
ejpam-5240	513	7	k. mendoza	PERSON
ejpam-5240	514	1	j. pure	PERSON
ejpam-5240	514	2	17	CARDINAL
ejpam-5240	514	3	3	CARDINAL
ejpam-5240	514	4	2024	CARDINAL
ejpam-5240	514	5	1779-1803	DATE
ejpam-5240	514	6	deg(a	PERSON
ejpam-5240	514	7	k k	PERSON
ejpam-5240	515	1	2 2	CARDINAL
ejpam-5240	515	2	1	CARDINAL
ejpam-5240	515	3	1	CARDINAL
ejpam-5240	516	1	1	CARDINAL
ejpam-5240	516	2	9	CARDINAL
ejpam-5240	517	1	1	CARDINAL
ejpam-5240	517	2	9	CARDINAL
ejpam-5240	519	1	0	CARDINAL
ejpam-5240	519	2	k ≤ n	PERSON
ejpam-5240	521	1	9	CARDINAL
ejpam-5240	523	1	2	CARDINAL
ejpam-5240	523	2	1	CARDINAL
ejpam-5240	524	1	9	CARDINAL
ejpam-5240	524	2	2	CARDINAL
ejpam-5240	526	1	1	CARDINAL
ejpam-5240	527	1	1	CARDINAL
ejpam-5240	527	2	9	CARDINAL
ejpam-5240	528	1	1	CARDINAL
ejpam-5240	528	2	deg(a	PERSON
ejpam-5240	529	1	1	CARDINAL
ejpam-5240	529	2	∈	ORG
ejpam-5240	531	1	10	CARDINAL
ejpam-5240	532	1	x3	GPE
ejpam-5240	532	2	1	CARDINAL
ejpam-5240	533	1	1	CARDINAL
ejpam-5240	533	2	4	CARDINAL
ejpam-5240	533	3	4	CARDINAL
ejpam-5240	535	1	4	CARDINAL
ejpam-5240	535	2	11	CARDINAL
ejpam-5240	536	1	∈	ORG
ejpam-5240	536	2	deg(a	PERSON
ejpam-5240	536	3	4	CARDINAL
ejpam-5240	536	4	1	CARDINAL
ejpam-5240	536	5	3	CARDINAL
ejpam-5240	538	1	4	CARDINAL
ejpam-5240	538	2	5	CARDINAL
ejpam-5240	540	1	m.e.	PERSON
ejpam-5240	540	2	pelagio	GPE
ejpam-5240	540	3	n. mame	PERSON
ejpam-5240	540	4	k. mendoza	PERSON
ejpam-5240	541	1	j. pure	PERSON
ejpam-5240	541	2	17	CARDINAL
ejpam-5240	541	3	3	CARDINAL
ejpam-5240	541	4	2024	CARDINAL
ejpam-5240	541	5	1779-1803 1796	DATE
ejpam-5240	541	6	11	CARDINAL
ejpam-5240	542	1	5.1	CARDINAL
ejpam-5240	543	1	two	CARDINAL
ejpam-5240	543	2	1 ≤ k ≤	QUANTITY
ejpam-5240	543	3	0	CARDINAL
ejpam-5240	544	1	1 ≤ k ≤	QUANTITY
ejpam-5240	544	2	2	CARDINAL
ejpam-5240	545	1	3	CARDINAL
ejpam-5240	546	1	sn	ORG
ejpam-5240	547	1	1 ≤ k ≤	QUANTITY
ejpam-5240	547	2	x(⌊n k	PERSON
ejpam-5240	547	3	x(⌊n k ⌋)k	PERSON
ejpam-5240	548	1	n k	PERSON
ejpam-5240	550	1	x(⌊n k	PERSON
ejpam-5240	550	2	x(⌊n k ⌋)k	PERSON
ejpam-5240	551	1	sn	ORG
ejpam-5240	551	2	∈	ORG
ejpam-5240	552	1	⊆	CARDINAL
ejpam-5240	552	2	⊆	CARDINAL
ejpam-5240	553	1	∈	ORG
ejpam-5240	558	1	sn	ORG
ejpam-5240	560	1	n k	PERSON
ejpam-5240	562	1	3	CARDINAL
ejpam-5240	562	2	one	CARDINAL
ejpam-5240	563	1	1 ≤ k ≤	QUANTITY
ejpam-5240	564	1	11	CARDINAL
ejpam-5240	565	1	s6	GPE
ejpam-5240	565	2	x3	GPE
ejpam-5240	565	3	x4	GPE
ejpam-5240	565	4	x5	GPE
ejpam-5240	565	5	2	CARDINAL
ejpam-5240	567	1	x5}	GPE
ejpam-5240	567	2	x3}	GPE
ejpam-5240	567	3	x5}	GPE
ejpam-5240	567	4	x3	GPE
ejpam-5240	567	5	x5}	GPE
ejpam-5240	567	6	x4	PERSON
ejpam-5240	567	7	x5}	GPE
ejpam-5240	567	8	x4	PERSON
ejpam-5240	567	9	12	CARDINAL
ejpam-5240	568	1	m.e. pelagio	ORG
ejpam-5240	568	2	n. mame	PERSON
ejpam-5240	568	3	k. mendoza	PERSON
ejpam-5240	569	1	j. pure	PERSON
ejpam-5240	569	2	17	CARDINAL
ejpam-5240	569	3	3	CARDINAL
ejpam-5240	569	4	2024	CARDINAL
ejpam-5240	569	5	1779-1803 1797	DATE
ejpam-5240	569	6	x4	PERSON
ejpam-5240	569	7	x4	PERSON
ejpam-5240	569	8	x3, x4	ORG
ejpam-5240	569	9	x3, x5	ORG
ejpam-5240	569	10	x4	PERSON
ejpam-5240	569	11	x4	PERSON
ejpam-5240	569	12	12	CARDINAL
ejpam-5240	570	1	⊆	CARDINAL
ejpam-5240	570	2	1 ≤	QUANTITY
ejpam-5240	570	3	1 ≤	CARDINAL
ejpam-5240	571	1	x3, x4}] /∈ e(gs(6,2	ORG
ejpam-5240	572	1	2	CARDINAL
ejpam-5240	573	1	t3	ORG
ejpam-5240	573	2	x3, x4}]	ORG
ejpam-5240	574	1	6 2	CARDINAL
ejpam-5240	574	2	3 and |t3| = 3	QUANTITY
ejpam-5240	575	1	3	CARDINAL
ejpam-5240	578	1	10	CARDINAL
ejpam-5240	578	2	1 ≤ k ≤	QUANTITY
ejpam-5240	578	3	2	CARDINAL
ejpam-5240	579	1	10	CARDINAL
ejpam-5240	580	1	sn	ORG
ejpam-5240	581	1	1 ≤ k ≤	QUANTITY
ejpam-5240	582	1	n k	PERSON
ejpam-5240	584	1	x(⌊n k	PERSON
ejpam-5240	585	1	k ⌋)k	ORG
ejpam-5240	586	1	3	CARDINAL
ejpam-5240	588	1	m.e	GPE
ejpam-5240	588	2	pelagio	GPE
ejpam-5240	588	3	n. mame	PERSON
ejpam-5240	588	4	k. mendoza	PERSON
ejpam-5240	589	1	j. pure	PERSON
ejpam-5240	589	2	17	CARDINAL
ejpam-5240	589	3	3	CARDINAL
ejpam-5240	589	4	2024	CARDINAL
ejpam-5240	589	5	1779-1803	DATE
ejpam-5240	592	1	n k	PERSON
ejpam-5240	595	1	w ⊆	PRODUCT
ejpam-5240	597	1	k. hence	PERSON
ejpam-5240	598	1	+ b.	PERSON
ejpam-5240	599	1	w	GPE
ejpam-5240	600	1	sn	ORG
ejpam-5240	601	1	̸=	ORG
ejpam-5240	601	2	w	ORG
ejpam-5240	602	1	n k	PERSON
ejpam-5240	603	1	12	CARDINAL
ejpam-5240	604	1	s6	PERSON
ejpam-5240	604	2	x3	GPE
ejpam-5240	604	3	x4	GPE
ejpam-5240	604	4	x5	GPE
ejpam-5240	604	5	2	CARDINAL
ejpam-5240	604	6	12	CARDINAL
ejpam-5240	605	1	11	DATE
ejpam-5240	605	2	3	CARDINAL
ejpam-5240	605	3	6 2	CARDINAL
ejpam-5240	606	1	2	CARDINAL
ejpam-5240	606	2	s6	GPE
ejpam-5240	609	1	3	CARDINAL
ejpam-5240	609	2	10	CARDINAL
ejpam-5240	609	3	2	CARDINAL
ejpam-5240	612	1	6 2	CARDINAL
ejpam-5240	612	2	3	CARDINAL
ejpam-5240	612	3	4	CARDINAL
ejpam-5240	612	4	1	CARDINAL
ejpam-5240	613	1	⊆	CARDINAL
ejpam-5240	615	1	10	CARDINAL
ejpam-5240	615	2	1	CARDINAL
ejpam-5240	616	1	9	CARDINAL
ejpam-5240	616	2	9	CARDINAL
ejpam-5240	618	1	1	CARDINAL
ejpam-5240	619	1	1 ≤ k ≤	QUANTITY
ejpam-5240	620	1	11	CARDINAL
ejpam-5240	620	2	k ≤ n. theorem	PERSON
ejpam-5240	620	3	11	CARDINAL
ejpam-5240	622	1	0	CARDINAL
ejpam-5240	622	2	k ≤ n	PERSON
ejpam-5240	623	1	1	CARDINAL
ejpam-5240	624	1	7	CARDINAL
ejpam-5240	625	1	1	CARDINAL
ejpam-5240	626	1	1	CARDINAL
ejpam-5240	626	2	m.e.	ORG
ejpam-5240	626	3	pelagio	GPE
ejpam-5240	626	4	n. mame	PERSON
ejpam-5240	626	5	k. mendoza	PERSON
ejpam-5240	627	1	j. pure	PERSON
ejpam-5240	627	2	17	CARDINAL
ejpam-5240	627	3	3	CARDINAL
ejpam-5240	627	4	2024	CARDINAL
ejpam-5240	627	5	1779-1803	DATE
ejpam-5240	627	6	5.2	CARDINAL
ejpam-5240	628	1	1 ≤ k ≤	QUANTITY
ejpam-5240	628	2	0	CARDINAL
ejpam-5240	629	1	1 ≤ k ≤	QUANTITY
ejpam-5240	629	2	2	CARDINAL
ejpam-5240	630	1	4	CARDINAL
ejpam-5240	632	1	1 ≤ k ≤	QUANTITY
ejpam-5240	632	2	x(⌊n k	PERSON
ejpam-5240	632	3	x(⌊n k ⌋)k	PERSON
ejpam-5240	632	4	n k	PERSON
ejpam-5240	634	1	x(⌊n k	PERSON
ejpam-5240	634	2	x(⌊n k ⌋)k	PERSON
ejpam-5240	635	1	sn	ORG
ejpam-5240	636	1	⊆	CARDINAL
ejpam-5240	636	2	⊆	CARDINAL
ejpam-5240	637	1	sn	ORG
ejpam-5240	638	1	∈	ORG
ejpam-5240	638	2	∈	ORG
ejpam-5240	638	3	7	DATE
ejpam-5240	643	1	+ b.	PERSON
ejpam-5240	644	1	sn	ORG
ejpam-5240	646	1	∈	ORG
ejpam-5240	646	2	∈	ORG
ejpam-5240	648	1	n k	PERSON
ejpam-5240	649	1	n k	PERSON
ejpam-5240	650	1	4	CARDINAL
ejpam-5240	652	1	1 ≤ k ≤	QUANTITY
ejpam-5240	653	1	13	CARDINAL
ejpam-5240	654	1	s6	GPE
ejpam-5240	654	2	x3	GPE
ejpam-5240	654	3	x4	GPE
ejpam-5240	654	4	x5	GPE
ejpam-5240	654	5	2	CARDINAL
ejpam-5240	654	6	12	CARDINAL
ejpam-5240	656	1	⊆	CARDINAL
ejpam-5240	658	1	x5}	GPE
ejpam-5240	658	2	x3}	GPE
ejpam-5240	658	3	x5}	GPE
ejpam-5240	658	4	x3	GPE
ejpam-5240	658	5	x5}	GPE
ejpam-5240	658	6	x4	PERSON
ejpam-5240	658	7	x5}	GPE
ejpam-5240	658	8	x4	PERSON
ejpam-5240	658	9	x5}	GPE
ejpam-5240	658	10	x3}	GPE
ejpam-5240	658	11	x5}	GPE
ejpam-5240	659	1	x3	GPE
ejpam-5240	659	2	x5}	GPE
ejpam-5240	659	3	x4	PERSON
ejpam-5240	659	4	x5}	GPE
ejpam-5240	659	5	x4	PERSON
ejpam-5240	664	1	3	CARDINAL
ejpam-5240	664	2	6 2	CARDINAL
ejpam-5240	666	1	6 2	CARDINAL
ejpam-5240	667	1	lemma 4	ORG
ejpam-5240	669	1	1 ≤ k ≤	QUANTITY
ejpam-5240	669	2	2	CARDINAL
ejpam-5240	670	1	m.e.	PERSON
ejpam-5240	670	2	pelagio	GPE
ejpam-5240	670	3	n. mame	PERSON
ejpam-5240	670	4	k. mendoza	PERSON
ejpam-5240	671	1	j. pure	PERSON
ejpam-5240	671	2	17	CARDINAL
ejpam-5240	671	3	3	CARDINAL
ejpam-5240	671	4	2024	CARDINAL
ejpam-5240	671	5	1779-1803 1800	DATE
ejpam-5240	671	6	12	CARDINAL
ejpam-5240	673	1	1 ≤ k ≤	QUANTITY
ejpam-5240	674	1	n k	PERSON
ejpam-5240	676	1	x(⌊n k	PERSON
ejpam-5240	677	1	k ⌋)k	ORG
ejpam-5240	678	1	lemma 4	ORG
ejpam-5240	683	1	n k	PERSON
ejpam-5240	686	1	w ⊆	PRODUCT
ejpam-5240	686	2	w	ORG
ejpam-5240	688	1	k.	PERSON
ejpam-5240	690	1	+ b.	PERSON
ejpam-5240	696	1	n k	PERSON
ejpam-5240	697	1	14	DATE
ejpam-5240	698	1	s6	GPE
ejpam-5240	698	2	x3	GPE
ejpam-5240	698	3	x4	GPE
ejpam-5240	698	4	x5	GPE
ejpam-5240	698	5	2	CARDINAL
ejpam-5240	699	1	13	CARDINAL
ejpam-5240	701	1	less than 3	CARDINAL
ejpam-5240	702	1	3	CARDINAL
ejpam-5240	702	2	12	CARDINAL
ejpam-5240	702	3	2	CARDINAL
ejpam-5240	705	1	6 2	CARDINAL
ejpam-5240	705	2	3	CARDINAL
ejpam-5240	705	3	4	CARDINAL
ejpam-5240	706	1	γ(gs(n,1	PERSON
ejpam-5240	707	1	12	CARDINAL
ejpam-5240	707	2	1	CARDINAL
ejpam-5240	707	3	γ(gs(n,1	PERSON
ejpam-5240	708	1	10	CARDINAL
ejpam-5240	709	1	10	CARDINAL
ejpam-5240	711	1	1	CARDINAL
ejpam-5240	711	2	13	CARDINAL
ejpam-5240	712	1	k ≤ n. theorem	PERSON
ejpam-5240	712	2	13	CARDINAL
ejpam-5240	714	1	0	CARDINAL
ejpam-5240	714	2	k ≤	PERSON
ejpam-5240	716	1	1	CARDINAL
ejpam-5240	717	1	7	DATE
ejpam-5240	718	1	1	CARDINAL
ejpam-5240	719	1	1	CARDINAL
ejpam-5240	719	2	m.e.	ORG
ejpam-5240	719	3	pelagio	GPE
ejpam-5240	719	4	n. mame	PERSON
ejpam-5240	719	5	k. mendoza	PERSON
ejpam-5240	720	1	j. pure	PERSON
ejpam-5240	720	2	17	CARDINAL
ejpam-5240	720	3	3	CARDINAL
ejpam-5240	720	4	2024	CARDINAL
ejpam-5240	720	5	1779-1803	DATE
ejpam-5240	720	6	5.3	CARDINAL
ejpam-5240	720	7	two	CARDINAL
ejpam-5240	720	8	1 ≤ k ≤	QUANTITY
ejpam-5240	720	9	0	CARDINAL
ejpam-5240	721	1	1 ≤ k ≤	QUANTITY
ejpam-5240	722	1	3	CARDINAL
ejpam-5240	722	2	lemma 4	ORG
ejpam-5240	723	1	14	DATE
ejpam-5240	724	1	sn	ORG
ejpam-5240	725	1	1 ≤ k ≤	QUANTITY
ejpam-5240	726	1	n k	PERSON
ejpam-5240	728	1	x(⌊n k	PERSON
ejpam-5240	729	1	k ⌋)k	ORG
ejpam-5240	730	1	3	CARDINAL
ejpam-5240	731	1	n k	PERSON
ejpam-5240	733	1	∈	ORG
ejpam-5240	733	2	deg(a	PERSON
ejpam-5240	733	3	0	CARDINAL
ejpam-5240	734	1	lemma 4	ORG
ejpam-5240	736	1	n k	PERSON
ejpam-5240	740	1	12	CARDINAL
ejpam-5240	741	1	n k	PERSON
ejpam-5240	746	1	n k	PERSON
ejpam-5240	747	1	15	CARDINAL
ejpam-5240	750	1	1	CARDINAL
ejpam-5240	751	1	7	CARDINAL
ejpam-5240	752	1	1	CARDINAL
ejpam-5240	754	1	1	CARDINAL
ejpam-5240	754	2	6	CARDINAL
ejpam-5240	755	1	two	CARDINAL
ejpam-5240	758	1	k = n.	PERSON
ejpam-5240	758	2	1	CARDINAL
ejpam-5240	758	3	∈	ORG
ejpam-5240	758	4	deg(a	PERSON
ejpam-5240	760	1	1 ≤ k ≤	QUANTITY
ejpam-5240	761	1	deg(a	PERSON
ejpam-5240	762	1	1	CARDINAL
ejpam-5240	763	1	2	CARDINAL
ejpam-5240	763	2	1 ≤ k ≤	QUANTITY
ejpam-5240	763	3	2	CARDINAL
ejpam-5240	763	4	1802	DATE
ejpam-5240	763	5	0	CARDINAL
ejpam-5240	763	6	3	CARDINAL
ejpam-5240	763	7	2	CARDINAL
ejpam-5240	765	1	∈	ORG
ejpam-5240	765	2	1 ≤ k ≤	QUANTITY
ejpam-5240	765	3	deg(a	PERSON
ejpam-5240	765	4	n−k k	PERSON
ejpam-5240	765	5	0	CARDINAL
ejpam-5240	765	6	k ≤ n	PERSON
ejpam-5240	765	7	deg(a	PERSON
ejpam-5240	766	1	0.	PERCENT
ejpam-5240	768	1	1 ≤ k ≤	QUANTITY
ejpam-5240	768	2	0	CARDINAL
ejpam-5240	768	3	k ≤ n	PERSON
ejpam-5240	768	4	1	CARDINAL
ejpam-5240	770	1	1 ≤ k ≤	QUANTITY
ejpam-5240	772	1	n k	PERSON
ejpam-5240	775	1	1	CARDINAL
ejpam-5240	781	1	batangas state university	ORG
ejpam-5240	781	2	the national engineering university	ORG
ejpam-5240	783	1	1	CARDINAL
ejpam-5240	783	2	a. neumaier a. brouwer	PERSON
ejpam-5240	783	3	a. cohen	PERSON
ejpam-5240	785	1	springer berlin	GPE
ejpam-5240	785	2	heidelberg	GPE
ejpam-5240	785	3	1989	DATE
ejpam-5240	786	1	1803	DATE
ejpam-5240	786	2	2	CARDINAL
ejpam-5240	786	3	richard a. brualdi	PERSON
ejpam-5240	788	1	prentice-hall	ORG
ejpam-5240	788	2	2010	DATE
ejpam-5240	789	1	3	CARDINAL
ejpam-5240	789	2	j. harris et al	PERSON
ejpam-5240	789	3	combinatorics	ORG
ejpam-5240	790	1	2008	DATE
ejpam-5240	791	1	4	CARDINAL
ejpam-5240	791	2	d. fowler	PERSON
ejpam-5240	793	1	american	NORP
ejpam-5240	793	2	monthly, 1996	DATE
ejpam-5240	794	1	5	CARDINAL
ejpam-5240	794	2	m. golumbic	PERSON
ejpam-5240	796	1	2004	DATE
ejpam-5240	797	1	6	CARDINAL
ejpam-5240	799	1	whitman college 2017,	ORG
ejpam-5240	799	2	2017	DATE
ejpam-5240	800	1	7	CARDINAL
ejpam-5240	800	2	i. hamid	PERSON
ejpam-5240	800	3	s. balamurugan	PERSON
ejpam-5240	802	1	arab journal of mathematical sciences	ORG
ejpam-5240	802	2	2016	DATE
ejpam-5240	803	1	8	CARDINAL
ejpam-5240	803	2	n. helwig	PERSON
ejpam-5240	805	1	university of minnesota	ORG
ejpam-5240	805	2	2020	DATE
ejpam-5240	806	1	9	CARDINAL
ejpam-5240	806	2	d. magpantay j.c. bonifacio	PERSON
ejpam-5240	806	3	c.j. andaya	GPE
ejpam-5240	808	1	european journal	ORG
ejpam-5240	808	2	2023	CARDINAL
ejpam-5240	809	1	10	CARDINAL
ejpam-5240	809	2	a. goodman	PERSON
ejpam-5240	811	1	canadian	NORP
ejpam-5240	811	2	18(1):106–112, 1966	DATE
ejpam-5240	812	1	11	CARDINAL
ejpam-5240	812	2	m. schaefer	PERSON
ejpam-5240	814	1	2010	DATE
ejpam-5240	815	1	12	CARDINAL
ejpam-5240	817	1	1993	DATE
ejpam-5240	818	1	13	CARDINAL
ejpam-5240	818	2	a. sugumaran	PERSON
ejpam-5240	818	3	e. jayachandran	PERSON
ejpam-5240	820	1	2018	DATE
ejpam-5240	821	1	14	CARDINAL
ejpam-5240	821	2	e. szpilrajn-marczewski	PERSON
ejpam-5240	822	1	two	CARDINAL
ejpam-5240	823	1	american mathematical society	ORG
ejpam-5240	823	2	33:303–307	CARDINAL
ejpam-5240	824	1	15	CARDINAL
ejpam-5240	824	2	t. mckee	PERSON
ejpam-5240	826	1	philadelphia	GPE
ejpam-5240	826	2	united states	GPE
ejpam-5240	826	3	1999	DATE