{
  "index": "1958-2-B-7",
  "type": "COMB",
  "tag": [
    "COMB",
    "NT"
  ],
  "difficulty": "",
  "question": "7. Let \\( a_{1}, a_{2}, \\ldots, a_{n} \\) be a permutation of the integers \\( 1,2, \\ldots, n \\). Call \\( a_{i} \\) a \"big\" integer if \\( a_{i}>a_{j} \\) for all \\( j>i \\). Find the mean number of \"big\" integers over all permutations on the first \\( n \\) positive integers.",
  "solution": "First Solution. If \\( \\sigma \\) is a permutation, let \\( N_{i}(\\sigma) \\) be the number of \"big\" integers occurring at position \\( i \\). Then \\( N_{i}(\\sigma)=0 \\) or 1 . The average value of \\( N_{i}(\\sigma) \\) over all the permutations is \\( 1 /(n-i+1) \\) because after \\( a_{1}, a_{2}, \\ldots \\), \\( a_{i-1} \\) have been selected, the question of whether or not \\( a_{i} \\) will be a big integer is whether or not \\( a_{i} \\) is the greatest among the ( \\( n-i+1 \\) ) integers left.\n\nLet the number of big integers in \\( \\sigma \\) be \\( N(\\sigma) \\). Then \\( N(\\sigma)=N_{1}(\\sigma)+N_{2}(\\sigma) \\) \\( +\\cdots+N_{n}(\\sigma) \\), and the average value of \\( N(\\sigma) \\) over all the \\( n \\) ! permutations will be the sum of the average values of the separate terms, \\( N_{i}(\\sigma) \\), for \\( i= \\) \\( 1,2, \\ldots, n \\). Hence this average is\n\\[\n1 / n+1 /(n-1)+1 /(n-2)+\\cdots+1\n\\]\n\nSecond Solution. The average number of big integers in a permutation of \\( n \\) distinct integers is the same no matter what these integers are. Call this number \\( A_{n} \\).\n\nGiven a permutation of \\( \\{1,2, \\ldots n\\} \\), remove the element 1 and close up to obtain a permutation of \\( \\{2,3, \\ldots, n\\} \\). This defines an \\( n \\)-to- 1 mapping \\( \\xi \\) of permutations of \\( \\{1,2, \\ldots, n\\} \\) into permutations of \\( \\{2,3, \\ldots, n\\} \\). If the element 1 appeared at the end of the original permutation \\( \\sigma \\), then \\( \\xi(\\sigma) \\) has one big integer fewer than \\( \\sigma \\). In all other cases \\( \\xi(\\sigma) \\) has the same number of big integers as \\( \\sigma \\). If \\( N(\\sigma) \\) denotes the number of big integers in \\( \\sigma \\). this means that\n\\[\nN(\\sigma)=N(\\xi(\\sigma))+1 \\quad \\text { for }(n-1)!\\text { permutations } \\sigma\n\\]\nand\n\\[\nN(\\sigma)=N(\\xi(\\sigma)) \\quad \\text { for the remaining permutations. }\n\\]\n\nHence\n\\[\n\\sum_{n} N(\\sigma)=\\sum_{n} N(\\xi(\\sigma))+(n-1)!.\n\\]\n\nDividing through by \\( n! \\) we see that\n\\[\nA_{n}=\\frac{1}{n!} \\sum_{\\sigma} N(\\xi(\\sigma))+1 / n\n\\]\n\nSince the mapping \\( \\xi \\) is always exactly \\( \\boldsymbol{n} \\)-to- 1 .\n\\[\n\\frac{1}{n!} \\sum_{n} N(\\xi(\\sigma))=A_{n-1}\n\\]\n\nThis gives the recurrence\n\\[\nA_{n}=A_{n-1}+1 / n .\n\\]\n\nSince \\( A_{1}=1 \\). it follows that\n\\[\nA_{n}=1+1 / 2+1 / 3+\\cdots+1 / n\n\\]",
  "vars": [
    "i",
    "j",
    "\\\\sigma",
    "\\\\xi",
    "a_i",
    "a_j",
    "a_1",
    "a_2",
    "a_n",
    "N",
    "N_i",
    "N_1",
    "N_2",
    "N_n"
  ],
  "params": [
    "n",
    "A",
    "A_n",
    "A_n-1",
    "A_1"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "i": "indexi",
        "j": "indexj",
        "\\\\sigma": "permute",
        "\\\\xi": "compress",
        "a_i": "elementi",
        "a_j": "elementj",
        "a_1": "elementone",
        "a_2": "elementtwo",
        "a_n": "elementlast",
        "N": "bigcount",
        "N_i": "bigcounti",
        "N_1": "bigcountone",
        "N_2": "bigcounttwo",
        "N_n": "bigcountlast",
        "n": "totalnum",
        "A": "avgcount",
        "A_n": "avgcountn",
        "A_n-1": "avgcountprev",
        "A_1": "avgcountone"
      },
      "question": "7. Let \\( elementone, elementtwo, \\ldots, elementlast \\) be a permutation of the integers \\( 1,2, \\ldots, totalnum \\). Call \\( elementi \\) a \"big\" integer if \\( elementi>elementj \\) for all \\( indexj>indexi \\). Find the mean number of \"big\" integers over all permutations on the first \\( totalnum \\) positive integers.",
      "solution": "First Solution. If \\( permute \\) is a permutation, let \\( bigcounti(permute) \\) be the number of \"big\" integers occurring at position \\( indexi \\). Then \\( bigcounti(permute)=0 \\) or 1. The average value of \\( bigcounti(permute) \\) over all the permutations is \\( 1 /(totalnum-indexi+1) \\) because after \\( elementone, elementtwo, \\ldots, elementi-1 \\) have been selected, the question of whether or not \\( elementi \\) will be a big integer is whether or not \\( elementi \\) is the greatest among the ( \\( totalnum-indexi+1 \\) ) integers left.\n\nLet the number of big integers in \\( permute \\) be \\( bigcount(permute) \\). Then \\( bigcount(permute)=bigcountone(permute)+bigcounttwo(permute)+\\cdots+bigcountlast(permute) \\), and the average value of \\( bigcount(permute) \\) over all the \\( totalnum ! \\) permutations will be the sum of the average values of the separate terms, \\( bigcounti(permute) \\), for \\( indexi =1,2, \\ldots, totalnum \\). Hence this average is\n\\[\n1 / totalnum+1 /(totalnum-1)+1 /(totalnum-2)+\\cdots+1\n\\]\n\nSecond Solution. The average number of big integers in a permutation of \\( totalnum \\) distinct integers is the same no matter what these integers are. Call this number \\( avgcount_{totalnum} \\).\n\nGiven a permutation of \\{1,2, \\ldots, totalnum\\}, remove the element 1 and close up to obtain a permutation of \\{2,3, \\ldots, totalnum\\}. This defines an \\( totalnum \\)-to-1 mapping \\( compress \\) of permutations of \\{1,2, \\ldots, totalnum\\} into permutations of \\{2,3, \\ldots, totalnum\\}. If the element 1 appeared at the end of the original permutation \\( permute \\), then \\( compress(permute) \\) has one big integer fewer than \\( permute \\). In all other cases \\( compress(permute) \\) has the same number of big integers as \\( permute \\). If \\( bigcount(permute) \\) denotes the number of big integers in \\( permute \\), this means that\n\\[\nbigcount(permute)=bigcount(compress(permute))+1 \\quad \\text { for }(totalnum-1)!\\text { permutations } permute\n\\]\nand\n\\[\nbigcount(permute)=bigcount(compress(permute)) \\quad \\text { for the remaining permutations. }\n\\]\n\nHence\n\\[\n\\sum_{totalnum} bigcount(permute)=\\sum_{totalnum} bigcount(compress(permute))+(totalnum-1)!.\n\\]\n\nDividing through by \\( totalnum! \\) we see that\n\\[\navgcount_{totalnum}=\\frac{1}{totalnum!} \\sum_{permute} bigcount(compress(permute))+1 / totalnum\n\\]\n\nSince the mapping \\( compress \\) is always exactly \\( \\boldsymbol{totalnum} \\)-to-1,\n\\[\n\\frac{1}{totalnum!} \\sum_{totalnum} bigcount(compress(permute))=avgcountprev\n\\]\n\nThis gives the recurrence\n\\[\navgcount_{totalnum}=avgcountprev+1 / totalnum .\n\\]\n\nSince \\( avgcountone=1 \\), it follows that\n\\[\navgcount_{totalnum}=1+1 / 2+1 / 3+\\cdots+1 / totalnum\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "i": "hydrangea",
        "j": "marigold",
        "\\\\sigma": "hummingbird",
        "\\\\xi": "alligator",
        "a_i": "butterscotch",
        "a_j": "gingerbread",
        "a_1": "raspberries",
        "a_2": "blueberries",
        "a_n": "strawberries",
        "N": "quarterback",
        "N_i": "chandelier",
        "N_1": "dragonfruit",
        "N_2": "watercress",
        "N_n": "butternut",
        "n": "crocodile",
        "A": "paperclips",
        "A_n": "windbreaker",
        "A_n-1": "snowblower",
        "A_1": "matchstick"
      },
      "question": "7. Let \\( raspberries, blueberries, \\ldots, strawberries \\) be a permutation of the integers \\( 1,2, \\ldots, crocodile \\). Call \\( butterscotch \\) a \"big\" integer if \\( butterscotch>gingerbread \\) for all \\( marigold>hydrangea \\). Find the mean number of \"big\" integers over all permutations on the first \\( crocodile \\) positive integers.",
      "solution": "First Solution. If \\( hummingbird \\) is a permutation, let \\( chandelier(hummingbird) \\) be the number of \"big\" integers occurring at position \\( hydrangea \\). Then \\( chandelier(hummingbird)=0 \\) or 1 . The average value of \\( chandelier(hummingbird) \\) over all the permutations is \\( 1 /(crocodile-hydrangea+1) \\) because after \\( raspberries, blueberries, \\ldots, \\), \\( a_{i-1} \\) have been selected, the question of whether or not \\( butterscotch \\) will be a big integer is whether or not \\( butterscotch \\) is the greatest among the ( \\( crocodile-hydrangea+1 \\) ) integers left.\n\nLet the number of big integers in \\( hummingbird \\) be \\( quarterback(hummingbird) \\). Then \\( quarterback(hummingbird)=dragonfruit(hummingbird)+watercress(hummingbird) +\\cdots+butternut(hummingbird) \\), and the average value of \\( quarterback(hummingbird) \\) over all the \\( crocodile \\) ! permutations will be the sum of the average values of the separate terms, \\( chandelier(hummingbird) \\), for \\( hydrangea= \\) \\( 1,2, \\ldots, crocodile \\). Hence this average is\n\\[\n1 / crocodile+1 /(crocodile-1)+1 /(crocodile-2)+\\cdots+1\n\\]\n\nSecond Solution. The average number of big integers in a permutation of \\( crocodile \\) distinct integers is the same no matter what these integers are. Call this number \\( windbreaker \\).\n\nGiven a permutation of \\( \\{1,2, \\ldots crocodile\\} \\), remove the element 1 and close up to obtain a permutation of \\( \\{2,3, \\ldots, crocodile\\} \\). This defines an \\( crocodile \\)-to- 1 mapping \\( alligator \\) of permutations of \\( \\{1,2, \\ldots, crocodile\\} \\) into permutations of \\( \\{2,3, \\ldots, crocodile\\} \\). If the element 1 appeared at the end of the original permutation \\( hummingbird \\), then \\( alligator(hummingbird) \\) has one big integer fewer than \\( hummingbird \\). In all other cases \\( alligator(hummingbird) \\) has the same number of big integers as \\( hummingbird \\). If \\( quarterback(hummingbird) \\) denotes the number of big integers in \\( hummingbird \\). this means that\n\\[\nquarterback(hummingbird)=quarterback(alligator(hummingbird))+1 \\quad \\text { for }(crocodile-1)!\\text { permutations } hummingbird\n\\]\nand\n\\[\nquarterback(hummingbird)=quarterback(alligator(hummingbird)) \\quad \\text { for the remaining permutations. }\n\\]\n\nHence\n\\[\n\\sum_{crocodile} quarterback(hummingbird)=\\sum_{crocodile} quarterback(alligator(hummingbird))+(crocodile-1)!.\n\\]\n\nDividing through by \\( crocodile! \\) we see that\n\\[\nwindbreaker=\\frac{1}{crocodile!} \\sum_{hummingbird} quarterback(alligator(hummingbird))+1 / crocodile\n\\]\n\nSince the mapping \\( alligator \\) is always exactly \\( \\boldsymbol{crocodile} \\)-to- 1 .\n\\[\n\\frac{1}{crocodile!} \\sum_{crocodile} quarterback(alligator(hummingbird))=snowblower\n\\]\n\nThis gives the recurrence\n\\[\nwindbreaker=snowblower+1 / crocodile .\n\\]\n\nSince \\( matchstick=1 \\). it follows that\n\\[\nwindbreaker=1+1 / 2+1 / 3+\\cdots+1 / crocodile\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "i": "aggregate",
        "j": "alltotal",
        "\\\\sigma": "steadiness",
        "\\\\xi": "stability",
        "a_i": "randomnum",
        "a_j": "differnum",
        "a_1": "lastentry",
        "a_2": "nextlast",
        "a_n": "beginning",
        "N": "smallcount",
        "N_i": "tinycount",
        "N_1": "minicount",
        "N_2": "microcount",
        "N_n": "nanocount",
        "n": "fixedsize",
        "A": "totalsum",
        "A_n": "totalsumn",
        "A_n-1": "totalsumprevious",
        "A_1": "totalsumone"
      },
      "question": "7. Let \\( lastentry, nextlast, \\ldots, beginning \\) be a permutation of the integers \\( 1,2, \\ldots, fixedsize \\). Call \\( randomnum \\) a \"big\" integer if \\( randomnum>differnum \\) for all \\( alltotal>aggregate \\). Find the mean number of \"big\" integers over all permutations on the first \\( fixedsize \\) positive integers.",
      "solution": "First Solution. If \\( steadiness \\) is a permutation, let \\( tinycount(steadiness) \\) be the number of \"big\" integers occurring at position \\( aggregate \\). Then \\( tinycount(steadiness)=0 \\) or 1. The average value of \\( tinycount(steadiness) \\) over all the permutations is \\( 1 /(fixedsize-aggregate+1) \\) because after \\( lastentry, nextlast, \\ldots, a_{aggregate-1} \\) have been selected, the question of whether or not \\( randomnum \\) will be a big integer is whether or not \\( randomnum \\) is the greatest among the ( \\( fixedsize-aggregate+1 \\) ) integers left.\n\nLet the number of big integers in \\( steadiness \\) be \\( smallcount(steadiness) \\). Then \\( smallcount(steadiness)=tinycount(steadiness)+tinycount(steadiness) +\\cdots+tinycount(steadiness) \\), and the average value of \\( smallcount(steadiness) \\) over all the \\( fixedsize! \\) permutations will be the sum of the average values of the separate terms, \\( tinycount(steadiness) \\), for \\( aggregate=1,2, \\ldots, fixedsize \\). Hence this average is\n\\[\n1 / fixedsize+1 /(fixedsize-1)+1 /(fixedsize-2)+\\cdots+1\n\\]\n\nSecond Solution. The average number of big integers in a permutation of \\( fixedsize \\) distinct integers is the same no matter what these integers are. Call this number \\( totalsumn \\).\n\nGiven a permutation of \\{1,2, \\ldots fixedsize\\}, remove the element 1 and close up to obtain a permutation of \\{2,3, \\ldots, fixedsize\\}. This defines an \\( fixedsize \\)-to-1 mapping \\( stability \\) of permutations of \\{1,2, \\ldots, fixedsize\\} into permutations of \\{2,3, \\ldots, fixedsize\\}. If the element 1 appeared at the end of the original permutation \\( steadiness \\), then \\( stability(steadiness) \\) has one big integer fewer than \\( steadiness \\). In all other cases \\( stability(steadiness) \\) has the same number of big integers as \\( steadiness \\). If \\( smallcount(steadiness) \\) denotes the number of big integers in \\( steadiness \\) this means that\n\\[\nsmallcount(steadiness)=smallcount(stability(steadiness))+1 \\quad \\text { for }(fixedsize-1)!\\text { permutations } steadiness\n\\]\nand\n\\[\nsmallcount(steadiness)=smallcount(stability(steadiness)) \\quad \\text { for the remaining permutations. }\n\\]\n\nHence\n\\[\n\\sum_{fixedsize} smallcount(steadiness)=\\sum_{fixedsize} smallcount(stability(steadiness))+(fixedsize-1)!.\n\\]\n\nDividing through by \\( fixedsize! \\) we see that\n\\[\n totalsumn=\\frac{1}{fixedsize!} \\sum_{steadiness} smallcount(stability(steadiness))+1 / fixedsize\n\\]\n\nSince the mapping \\( stability \\) is always exactly \\( \\boldsymbol{fixedsize} \\)-to-1 .\n\\[\n\\frac{1}{fixedsize!} \\sum_{fixedsize} smallcount(stability(steadiness))=totalsumprevious\n\\]\n\nThis gives the recurrence\n\\[\n totalsumn=totalsumprevious+1 / fixedsize .\n\\]\n\nSince \\( totalsumone=1 \\). it follows that\n\\[\n totalsumn=1+1 / 2+1 / 3+\\cdots+1 / fixedsize\n\\]"
    },
    "garbled_string": {
      "map": {
        "i": "qzxwvtnp",
        "j": "hjgrksla",
        "\\\\sigma": "bvlacmker",
        "\\\\xi": "ruqsonpey",
        "a_i": "xiejahtru",
        "a_j": "nquoptyel",
        "a_1": "sordkfynm",
        "a_2": "vclimgepr",
        "a_n": "kwjzfehio",
        "N": "tdufhplok",
        "N_i": "wamnzcbrt",
        "N_1": "gepluvhqa",
        "N_2": "mobqtrzen",
        "N_n": "yfsckdlar",
        "n": "pjroxdiae",
        "A": "hvdslkmab",
        "A_n": "zufqlenmx",
        "A_n-1": "vkptaejro",
        "A_1": "dhpqxirsm"
      },
      "question": "7. Let \\( sordkfynm, vclimgepr, \\ldots, kwjzfehio \\) be a permutation of the integers \\( 1,2, \\ldots, pjroxdiae \\). Call \\( xiejahtru \\) a \"big\" integer if \\( xiejahtru>nquoptyel \\) for all \\( hjgrksla>qzxwvtnp \\). Find the mean number of \"big\" integers over all permutations on the first \\( pjroxdiae \\) positive integers.",
      "solution": "First Solution. If \\( bvlacmker \\) is a permutation, let \\( wamnzcbrt(bvlacmker) \\) be the number of \"big\" integers occurring at position \\( qzxwvtnp \\). Then \\( wamnzcbrt(bvlacmker)=0 \\) or 1. The average value of \\( wamnzcbrt(bvlacmker) \\) over all the permutations is \\( 1 /(pjroxdiae-qzxwvtnp+1) \\) because after \\( sordkfynm, vclimgepr, \\ldots, a_{qzxwvtnp-1} \\) have been selected, the question of whether or not \\( xiejahtru \\) will be a big integer is whether or not \\( xiejahtru \\) is the greatest among the ( \\( pjroxdiae-qzxwvtnp+1 \\) ) integers left.\n\nLet the number of big integers in \\( bvlacmker \\) be \\( tdufhplok(bvlacmker) \\). Then \\( tdufhplok(bvlacmker)=gepluvhqa(bvlacmker)+mobqtrzen(bvlacmker)+\\cdots+yfsckdlar(bvlacmker) \\), and the average value of \\( tdufhplok(bvlacmker) \\) over all the \\( pjroxdiae \\)! permutations will be the sum of the average values of the separate terms, \\( wamnzcbrt(bvlacmker) \\), for \\( qzxwvtnp=1,2, \\ldots, pjroxdiae \\). Hence this average is\n\\[\n1 / pjroxdiae+1 /(pjroxdiae-1)+1 /(pjroxdiae-2)+\\cdots+1\n\\]\n\nSecond Solution. The average number of big integers in a permutation of \\( pjroxdiae \\) distinct integers is the same no matter what these integers are. Call this number \\( zufqlenmx \\).\n\nGiven a permutation of \\( \\{1,2, \\ldots pjroxdiae\\} \\), remove the element 1 and close up to obtain a permutation of \\( \\{2,3, \\ldots, pjroxdiae\\} \\). This defines an \\( pjroxdiae \\)-to-1 mapping \\( ruqsonpey \\) of permutations of \\{1,2, \\ldots, pjroxdiae\\} into permutations of \\{2,3, \\ldots, pjroxdiae\\}. If the element 1 appeared at the end of the original permutation \\( bvlacmker \\), then \\( ruqsonpey(bvlacmker) \\) has one big integer fewer than \\( bvlacmker \\). In all other cases \\( ruqsonpey(bvlacmker) \\) has the same number of big integers as \\( bvlacmker \\). If \\( tdufhplok(bvlacmker) \\) denotes the number of big integers in \\( bvlacmker \\), this means that\n\\[\ntdufhplok(bvlacmker)=tdufhplok(ruqsonpey(bvlacmker))+1 \\quad \\text { for }(pjroxdiae-1)!\\text { permutations } bvlacmker\n\\]\nand\n\\[\ntdufhplok(bvlacmker)=tdufhplok(ruqsonpey(bvlacmker)) \\quad \\text { for the remaining permutations. }\n\\]\n\nHence\n\\[\n\\sum_{pjroxdiae} tdufhplok(bvlacmker)=\\sum_{pjroxdiae} tdufhplok(ruqsonpey(bvlacmker))+(pjroxdiae-1)!.\n\\]\n\nDividing through by \\( pjroxdiae! \\) we see that\n\\[\nzufqlenmx=\\frac{1}{pjroxdiae!} \\sum_{bvlacmker} tdufhplok(ruqsonpey(bvlacmker))+1 / pjroxdiae\n\\]\n\nSince the mapping \\( ruqsonpey \\) is always exactly \\( \\mathbf{pjroxdiae} \\)-to-1,\n\\[\n\\frac{1}{pjroxdiae!} \\sum_{pjroxdiae} tdufhplok(ruqsonpey(bvlacmker))=vkptaejro\n\\]\n\nThis gives the recurrence\n\\[\nzufqlenmx=vkptaejro+1 / pjroxdiae .\n\\]\n\nSince \\( dhpqxirsm=1 \\), it follows that\n\\[\nzufqlenmx=1+1 / 2+1 / 3+\\cdots+1 / pjroxdiae\n\\]"
    },
    "kernel_variant": {
      "question": "Let $d\\ge 2$ be fixed.  On the $d$–dimensional cubic lattice  \n\\[\n[n]^{d}:=\\Bigl\\{(i_{1},\\dots ,i_{d}):1\\le i_{k}\\le n\\;(k=1,\\dots ,d)\\Bigr\\}\n\\]\nplace the $N:=n^{d}$ distinct “masses’’ $1,2,\\dots ,N$ by means of a\nuniformly random bijection  \n\\[\nm:[n]^{d}\\longrightarrow\\{1,2,\\dots ,N\\}.\n\\]\n\nFor $v=(v_{1},\\dots ,v_{d})$ put  \n\\[\nQ(v):=\\Bigl\\{w\\in[n]^{d}:w_{k}\\ge v_{k}\\;(1\\le k\\le d)\\Bigr\\},\n\\qquad\ns(v):=|Q(v)|=\\prod_{k=1}^{d}(n-v_{k}+1).\n\\]\n\nA vertex $v$ is said to be \\emph{deep–light} if  \n\\[\nm(v)<m(w)\\qquad(\\forall w\\in Q(v)\\setminus\\{v\\}).\n\\]\n\nDefine  \n\\[\nX_{n,d}:=\\#\\Bigl\\{v\\in[n]^{d}:v\\text{ is deep--light}\\Bigr\\},\n\\qquad\nH_{n}:=\\sum_{k=1}^{n}\\frac{1}{k}.\n\\]\n\n1.  Prove that  \n\\[\n\\mathbb{E}[X_{n,d}]=(H_{n})^{d}.\n\\]\n\n2.  Show that there exists a constant $C_{d}<\\infty$, depending only on\n$d$, such that  \n\\[\n\\operatorname{Var}\\!\\bigl(X_{n,d}\\bigr)\\le C_{d}\\,(H_{n})^{\\,2d}. \\tag{$\\dagger$}\n\\]\n\n3.  Deduce from $(\\dagger)$ that the sequence  \n\\[\nY_{n,d}:=\\frac{X_{n,d}}{(H_{n})^{\\,d}}\n\\]\nis \\emph{uniformly $L^{2}$–bounded}.  Conclude, in particular, that the\nfamily $\\{Y_{n,d}:n\\ge 1\\}$ is tight.\n\n(The standard estimates $H_{n}=\\log n+\\gamma+O(n^{-1})$ and\n$\\displaystyle\\sum_{k>t}k^{-2}\\le t^{-1}$ may be used without proof.)\n\n--------------------------------------------------------------------",
      "solution": "Throughout write  \n\\[\nN:=n^{d},\\qquad\nI_{v}:=\\mathbf 1_{\\{v\\text{ is deep--light}\\}},\\qquad\nX_{n,d}=\\sum_{v\\in[n]^{d}} I_{v}.\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n1.\\;Expectation\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nFix $v\\in[n]^{d}$.  Inside $Q(v)$ the restriction\n$m|_{Q(v)}$ is a uniformly random permutation of the $s(v)$ masses\nlocated there.  Every one of the $s(v)$ sites therefore\nreceives the smallest mass in $Q(v)$ with probability $1/s(v)$, whence  \n\\[\n\\mathbb{P}(I_{v}=1)=\\frac{1}{s(v)}\n\\quad\\Longrightarrow\\quad\n\\mathbb{E}[X_{n,d}]\n=\\sum_{v\\in[n]^{d}}\\frac{1}{s(v)}\n      =\\biggl(\\sum_{t=1}^{n}\\frac{1}{t}\\biggr)^{d}\n      =(H_{n})^{d}.\n\\]\n\\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n2.\\;Variance upper bound\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWrite  \n\\[\nA:=Q(u),\\quad a:=|A|=s(u),\\qquad\nB:=Q(v),\\quad b:=|B|=s(v).\n\\]\nSet $S:=A\\cup B$ and $s:=|S|$.\n\n----------------------------------------------------------------\n2A.\\;A universal covariance bound\n----------------------------------------------------------------\nFix \\emph{distinct} vertices $u,v$ and assume without loss of\ngenerality that $a\\le b$.\nWe establish\n\\[\n\\bigl|\\operatorname{Cov}(I_{u},I_{v})\\bigr|\n      \\le\\frac{C}{ab}\\qquad(\\text{with }C=3). \\tag{2.1}\n\\]\n\n\\emph{Proof of (2.1).}\nBecause\n$\\operatorname{Cov}(I_{u},I_{v})\n      =\\mathbb{P}(I_{u}=I_{v}=1)-\\dfrac{1}{ab}$,\nit suffices to show\n\\[\n\\mathbb{P}(I_{u}=I_{v}=1)\\le\\frac{2}{ab}. \\tag{2.2}\n\\]\n\nReveal first the relative order of the $b$ masses inside $B=Q(v)$.\nAmong the $b!$ equiprobable orders, $v$ carries the smallest mass in\nexactly $(b-1)!$ of them, hence\n\\[\n\\mathbb{P}\\bigl(I_{v}=1\\bigr)=\\frac{1}{b}.\n\\]\n\nCondition on the event $I_{v}=1$.\nTwo cases occur.\n\n(i) $u\\notin B$.  \nThen \\emph{no} information has been revealed about the ordering of the\nmasses inside $A=Q(u)$, whence\n\\[\n\\mathbb{P}\\bigl(I_{u}=1\\,\\bigm|\\,I_{v}=1\\bigr)=\\frac{1}{a}.\n\\]\n\n(ii) $u\\in B$.  \nSince $v$ is now known to be the smallest in $B$, the conditional\ndistribution of the remaining $b-1$ masses in $B\\setminus\\{v\\}$ is\nstill uniform.  Inside $A=Q(u)$ the vertex $v$ is \\emph{not} present,\nso $u$ must beat the remaining $a-1$ points of $A\\setminus\\{u\\}$.\nConsequently\n\\[\n\\mathbb{P}\\bigl(I_{u}=1\\,\\bigm|\\,I_{v}=1\\bigr)\n      \\le\\frac{1}{a-1}\\le\\frac{2}{a},\n      \\qquad\\text{since }a\\ge2.\n\\]\n\nIn both cases\n$\\mathbb{P}(I_{u}=1\\,\\&\\,I_{v}=1)\\le\\dfrac{2}{ab}$, proving (2.2).\nCombining (2.2) with $\\mathbb{P}(I_{u}=1)\\mathbb{P}(I_{v}=1)=1/(ab)$\nyields (2.1).\n\\hfill$\\square$\n\n----------------------------------------------------------------\n2B.\\;Arithmetic preliminaries\n----------------------------------------------------------------\nIntroduce\n\\[\n\\Sigma_{d}(t):=\\#\\Bigl\\{(i_{1},\\dots ,i_{d})\\in\\mathbb{N}^{d}:\n                      i_{1}\\cdots i_{d}\\le t\\Bigr\\},\n\\qquad\n\\Delta_{d}(t):=\\Sigma_{d}(t)-\\Sigma_{d}(t-1)\n              =\\#\\Bigl\\{(i_{1},\\dots ,i_{d}):\n                      i_{1}\\cdots i_{d}=t\\Bigr\\}.\n\\]\n\nA classic divisor–counting estimate (cf.\\ Hardy–Wright, Chap.\\;V)\nimplies\n\\[\n\\Sigma_{d}(t)\\le C_{d}\\,t(\\log t)^{d-1}\\quad(t\\ge2), \\tag{2.3}\n\\]\nhence\n\\[\n\\Delta_{d}(t)\\le C_{d}\\,(\\log t)^{\\,d-1}\\quad(t\\ge3). \\tag{2.4}\n\\]\n\nMoreover, for every $\\alpha>0$ and $M\\ge3$,\n\\[\n\\sum_{t=1}^{M}\\frac{(\\log t)^{\\alpha}}{t}\n  \\le C_{\\alpha}\\,(\\log M)^{\\alpha+1}. \\tag{2.5}\n\\]\n\n----------------------------------------------------------------\n2C.\\;Bounding the double sum\n----------------------------------------------------------------\nDecompose\n\\[\n\\operatorname{Var}(X_{n,d})\n=\\sum_{v}\\operatorname{Var}(I_{v})\n+2\\sum_{u<v}\\operatorname{Cov}(I_{u},I_{v}).  \\tag{2.6}\n\\]\n\n\\emph{Diagonal part.}  Since $0\\le I_{v}\\le1$,\n\\[\n\\sum_{v}\\operatorname{Var}(I_{v})\n       \\le\\sum_{v}\\mathbb{E}[I_{v}]\n       =(H_{n})^{d}.                                     \\tag{2.7}\n\\]\n\n\\emph{Mixed part.}  Order the pair $(u,v)$ so that $a=s(u)\\le b=s(v)$.\nFrom (2.1),\n\\[\n\\bigl|\\operatorname{Cov}(I_{u},I_{v})\\bigr|\n      \\le\\frac{3}{ab}=\\frac{3}{b}\\cdot\\frac{1}{a}.       \\tag{2.8}\n\\]\n\nFix $v$ and set\n\\[\nW_{v}:=\\sum_{\\substack{u\\neq v\\\\ a\\le b}}\\frac{1}{a}\n      =\\sum_{u\\neq v}\\mathbf 1_{\\{s(u)\\le s(v)\\}}\\frac{1}{s(u)}.\n\\]\nReplacing $i_{k}:=n-u_{k}+1$ provides a bijection from the set of $u$\ncounted in $W_{v}$ to  \n\\[\n\\Bigl\\{(i_{1},\\dots ,i_{d})\\in\\mathbb{N}^{d}:\n      i_{1}\\cdots i_{d}\\le b\\Bigr\\},\n\\]\nso that, using (2.3) and (2.5) with $\\alpha=d-1$,\n\\[\nW_{v}\\le\\sum_{t=1}^{b}\\frac{\\Delta_{d}(t)}{t}\n      \\le C_{d}\\sum_{t=3}^{b}\\frac{(\\log t)^{d-1}}{t}\n      \\le C_{d}\\,(\\log b)^{d}.                           \\tag{2.9}\n\\]\n\nHence\n\\[\n\\sum_{u<v}\\bigl|\\operatorname{Cov}(I_{u},I_{v})\\bigr|\n \\le 3\\sum_{v}\\frac{W_{v}}{b}\n \\le 3C_{d}\\sum_{v}\\frac{(\\log s(v))^{d}}{s(v)}.         \\tag{2.10}\n\\]\n\nGroup vertices according to the value $t=s(v)$ and apply (2.4):\n\\[\n\\sum_{v}\\frac{(\\log s(v))^{d}}{s(v)}\n  =\\sum_{t=1}^{n^{d}}\\frac{(\\log t)^{d}}{t}\\,\\Delta_{d}(t)\n  \\le C_{d}\\sum_{t=3}^{n^{d}}\\frac{(\\log t)^{2d-1}}{t}.  \\tag{2.11}\n\\]\n\nFinally, (2.5) with $\\alpha=2d-1$ gives\n\\[\n\\sum_{t=3}^{n^{d}}\\frac{(\\log t)^{2d-1}}{t}\n\\le C_{d}\\,(\\log n^{d})^{2d}\n = C_{d}\\,(d\\log n)^{2d}\n \\le C_{d}\\,(H_{n})^{2d}.                              \\tag{2.12}\n\\]\n\nCombining (2.6)–(2.12),\n\\[\n\\operatorname{Var}(X_{n,d})\n \\le (H_{n})^{d}+C_{d}'(H_{n})^{2d}\n \\le C_{d}''(H_{n})^{2d}\\qquad(n\\ge2),\n\\]\nestablishing $(\\dagger)$.\n\\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n3.\\;Uniform $L^{2}$–boundedness\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nSince $\\mathbb{E}[Y_{n,d}]=1$ and by $(\\dagger)$\n\\[\n\\operatorname{Var}(Y_{n,d})\n      =\\frac{\\operatorname{Var}(X_{n,d})}{(H_{n})^{2d}}\n      \\le\\frac{C_{d}}{1}=C_{d},\n\\]\nthe family $\\{Y_{n,d}\\}_{n\\ge1}$ is bounded in $L^{2}$ and therefore\ntight.\n\\hfill$\\square$\n\n----------------------------------------------------------------\nReferences  \n\n[1] G.\\;H. Hardy and E.\\;M. Wright, \\emph{An Introduction to the Theory\nof Numbers}, 6th ed., Oxford Univ.\\;Press, 2008.\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.501144",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimension:  the problem generalises the 1-dimensional\n   “record’’ concept to an arbitrary fixed number d ≥ 2 of dimensions,\n   increasing the state space from n to nᵈ sites.\n\n2. Additional constraints:  a site must be lighter than every crate in\n   a d-dimensional positive orthant, a much larger and overlapping set\n   of competitors than in the 1-dimensional case.\n\n3. Deeper theory:  \n   – Multi–index harmonic sums appear; separating variables is\n     essential to compute the expectation.  \n   – To bound the variance one must control covariances across a highly\n     inter-dependent family of indicator variables, requiring a careful\n     classification into comparable and incomparable pairs and an\n     estimate that hinges on geometric properties of the lattice.  \n   – Probabilistic limit results (Chebyshev together with (∗)) are\n     needed to prove convergence in probability.\n\n4. Multiple interacting concepts:  the solution blends enumerative\n   combinatorics (orthant sizes), multi–dimensional harmonic–analysis,\n   dependence estimates, and probabilistic limit theory.\n\n5. Greater length and depth:  compared with the single–sum harmonic\n   computation of the original problem, the enhanced variant demands\n   a multi–sum analysis, careful covariance bounds, and an explicit\n   asymptotic argument, each of which is several technical steps\n   longer than the entire original solution."
      }
    },
    "original_kernel_variant": {
      "question": "Let $d\\ge 2$ be fixed.  \nOn the $d$–dimensional cubic lattice  \n\\[\n[n]^{d}:=\\Bigl\\{(i_{1},\\dots ,i_{d}) : 1\\le i_{k}\\le n\\;(k=1,\\dots ,d)\\Bigr\\}\n\\]\nplace the $N:=n^{d}$ distinct “masses’’ $1,2,\\dots ,N$ by a uniformly\nrandom bijection  \n\\[\nm:[n]^{d}\\longrightarrow\\{1,2,\\dots ,N\\}.\n\\]\n\nFor $v=(v_{1},\\dots ,v_{d})$ put  \n\\[\nQ(v):=\\Bigl\\{w\\in[n]^{d}: w_{k}\\ge v_{k}\\;(1\\le k\\le d)\\Bigr\\},\n\\qquad\ns(v):=|Q(v)|=\\prod_{k=1}^{d}(n-v_{k}+1).\n\\]\n\nA vertex $v$ is called \\emph{deep–light} if  \n\\[\nm(v)<m(w)\\qquad(\\forall w\\in Q(v)\\setminus\\{v\\}).\n\\]\n\nWrite  \n\\[\nX_{n,d}:=\\#\\Bigl\\{v\\in[n]^{d}:v\\text{ is deep--light}\\Bigr\\},\n\\qquad\nH_{n}:=\\sum_{k=1}^{n}\\frac{1}{k}.\n\\]\n\n1.  Prove that  \n\\[\n\\mathbb{E}\\bigl[X_{n,d}\\bigr]=\\bigl(H_{n}\\bigr)^{d}.\n\\]\n\n2.  Show that there is a finite constant $C_{d}$, depending only on\n$d$, such that  \n\\[\n\\operatorname{Var}(X_{n,d})\\le C_{d}\\,\\bigl(H_{n}\\bigr)^{\\,2d-1}. \\tag{$\\dagger$}\n\\]\n\n3.  Deduce from $(\\dagger)$ that  \n\\[\n\\frac{X_{n,d}}{\\bigl(H_{n}\\bigr)^{\\,d}}\n          \\xrightarrow[n\\to\\infty]{L^{2}}1\n\\quad\\Longrightarrow\\quad\n\\frac{X_{n,d}}{\\bigl(H_{n}\\bigr)^{\\,d}}\n          \\xrightarrow[n\\to\\infty]{\\mathbb{P}}1 .\n\\]\n\n(The estimates $H_{n}=\\log n+\\gamma+O(n^{-1})$ and\n$\\displaystyle\\sum_{k>t}k^{-2}\\le t^{-1}$ may be used without proof.)\n\n-----------------------------------------------------------------",
      "solution": "Throughout write  \n\\[\nN:=n^{d},\\qquad\nI_{v}:=\\mathbf 1_{\\{v\\text{ is deep--light}\\}},\\qquad\nX_{n,d}=\\sum_{v\\in[n]^{d}} I_{v},\n\\quad\\text{and}\\quad\nu<v\\;:\\Longleftrightarrow\\;u\\neq v\\text{ and the ordered pair is taken once. }\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \n1.  The expectation  \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \n\nFor a fixed vertex $v$ the restriction of $m$ to\nthe orthant $Q(v)$ is a uniformly random permutation of the\n$s(v)$ masses located there.  Each site in $Q(v)$ therefore\nreceives the lightest of those $s(v)$ masses with probability\n$1/s(v)$, whence  \n\\[\n\\mathbb{P}(I_{v}=1)=\\frac{1}{s(v)}.\n\\]\nConsequently  \n\\[\n\\mathbb{E}\\bigl[X_{n,d}\\bigr]\n      =\\sum_{v\\in[n]^{d}}\\frac{1}{s(v)}\n      =\\Bigl(\\sum_{t=1}^{n}\\frac{1}{t}\\Bigr)^{d}\n      =(H_{n})^{d}.\n\\]\n\\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \n2.  An upper bound for $\\operatorname{Var}(X_{n,d})$  \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \n\nWe first establish a universal bound for  \n$\\lvert\\operatorname{Cov}(I_{u},I_{v})\\rvert$ and then show that the\nresulting double sum is  \n$O\\!\\bigl((H_{n})^{2d-1}\\bigr)$.\n\n----------------------------------------------------------------  \n2A.  A precise covariance bound  \n----------------------------------------------------------------  \n\nFix two distinct vertices $u\\neq v$ and put  \n\\[\nA:=Q(u),\\qquad B:=Q(v),\\qquad\na:=|A|,\\quad b:=|B|.\n\\]\n\n(i)  \\emph{Nested orthants ($A\\subset B$ or $B\\subset A$).}  \nAssume, for definiteness, $A\\subset B$.  \nBecause $v\\notin A$ (the strict inclusion forces $v$ to lie outside\n$A$ while $u\\in B$), the events  \n\\[\nI_{v}=1\\; \\Longleftrightarrow\\; m(v)=\\min_{B}m,\\qquad\nI_{u}=1\\; \\Longleftrightarrow\\; m(u)=\\min_{A}m\n\\]\nare \\emph{independent}.  \nIndeed, after conditioning on $I_{v}=1$ the smallest mass in $B$\nis fixed at $v$ and the remaining $b-1$ masses are still arranged\nuniformly at random, so the conditional probability that $u$ is the\nminimum in $A$ is exactly $1/a$.  Thus  \n\\[\n\\mathbb{P}(I_{u}=I_{v}=1)=\\frac{1}{a\\,b}\n\\quad\\Longrightarrow\\quad\n\\operatorname{Cov}(I_{u},I_{v})=0.\n\\]\n\n(ii)  \\emph{Non–nested orthants ($A\\not\\subset B$ and $B\\not\\subset\nA$).}  \nWrite\n\\[\ns:=|A\\cup B|=a+b-r,\\qquad r:=|A\\cap B|.\n\\]\nBecause $u\\notin B$ and $v\\notin A$, the two events $I_{u}=1$ and\n$I_{v}=1$ require that the two smallest masses inside $A\\cup B$ fall on\n$u$ and $v$ (in either order).  All $s$ positions are equally likely,\nso  \n\\[\n\\mathbb{P}(I_{u}=I_{v}=1)=\\frac{2}{s(s-1)}\n                       \\le\\frac{2}{\\max\\{a,b\\}^{2}}.\n\\]\n\nCombining (i)–(ii) we obtain the \\emph{uniform} bound  \n\\[\n\\lvert\\operatorname{Cov}(I_{u},I_{v})\\rvert\n      \\le\\frac{2}{\\max\\{a,b\\}^{2}}\n      \\qquad(u\\neq v). \\tag{2.1}\n\\]\n\n----------------------------------------------------------------  \n2B.  A summation lemma  \n----------------------------------------------------------------  \n\nFor $t\\ge 1$ put  \n\\[\n\\Sigma_{d}(t):=\\#\\Bigl\\{(i_{1},\\dots ,i_{d})\\in\\mathbb{N}^{d}:\n                      i_{1}\\cdots i_{d}\\le t\\Bigr\\}.\n\\]\nA standard divisor–counting argument (see\nHardy–Wright, Chap.~V) yields  \n\\[\n\\Sigma_{d}(t)\\le C_{d}\\,t\\,\\bigl(\\log t\\bigr)^{d-1}, \\qquad t\\ge 2, \\tag{2.2}\n\\]\nfor some constant $C_{d}$ depending only on $d$.  By partial\nsummation,\n\\[\n\\sum_{s\\le t}s\n      \\le C_{d}\\,t^{2}\\,(\\log t)^{d-1},               \\tag{2.3}\n\\qquad\n\\sum_{s\\le t}\\frac{1}{s}\n      \\le C_{d}\\,(\\log t)^{d}.                        \\tag{2.4}\n\\]\n\n----------------------------------------------------------------  \n2C.  Bounding the double sum of covariances  \n----------------------------------------------------------------  \n\nWrite the variance as\n\\[\n\\operatorname{Var}(X_{n,d})\n   =\\sum_{v}\\operatorname{Var}(I_{v})\n    +2\\sum_{u<v}\\operatorname{Cov}(I_{u},I_{v}). \\tag{2.5}\n\\]\n\nThe diagonal contribution satisfies\n\\[\n\\sum_{v}\\operatorname{Var}(I_{v})\n       \\le\\sum_{v}\\mathbb{E}[I_{v}]\n       =(H_{n})^{d}.                                   \\tag{2.6}\n\\]\n\nTurn to the mixed terms.  Order the vertices so that\n$s(u)\\le s(v)$.  From (2.1) we get  \n\\[\n\\bigl|\\operatorname{Cov}(I_{u},I_{v})\\bigr|\n      \\le\\frac{2}{s(v)^{2}}\\qquad(u\\neq v).            \\tag{2.7}\n\\]\nHence  \n\\[\n\\sum_{u<v}\\lvert\\operatorname{Cov}(I_{u},I_{v})\\rvert\n \\le 2\\sum_{v}\\frac{1}{s(v)^{2}}\n       \\sum_{\\substack{u:u\\neq v\\\\ s(u)\\le s(v)}}\\!s(u). \\tag{2.8}\n\\]\n\nFor the inner sum we invoke (2.3) with $t=s(v)$:\n\\[\n\\sum_{\\substack{u:u\\neq v\\\\ s(u)\\le s(v)}}\\!s(u)\n       \\le C_{d}\\,s(v)\\,\\bigl(\\log s(v)\\bigr)^{d-1}.   \\tag{2.9}\n\\]\n\nCombine (2.8)–(2.9) and use (2.4):\n\\[\n\\sum_{u<v}\\lvert\\operatorname{Cov}(I_{u},I_{v})\\rvert\n \\le 2C_{d}\\sum_{v}\\frac{(\\log s(v))^{d-1}}{s(v)}\n \\le 2C_{d}\\bigl(H_{n}\\bigr)^{2d-1}.                   \\tag{2.10}\n\\]\n\nFinally, gather (2.6) and (2.10):\n\\[\n\\operatorname{Var}(X_{n,d})\n      \\le (H_{n})^{d}+2C_{d}(H_{n})^{2d-1}\n      \\le C_{d}'(H_{n})^{2d-1}\\qquad(n\\ge 2),         \\tag{2.11}\n\\]\nwhere $C_{d}'=1+2C_{d}$.  This proves $(\\dagger)$.\n\\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \n3.  $L^{2}$–law of large numbers  \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  \n\nBy Parts 1–2,  \n\\[\n\\mathbb{E}\\!\\Bigl[\\bigl(X_{n,d}-(H_{n})^{d}\\bigr)^{2}\\Bigr]\n      =\\operatorname{Var}(X_{n,d})\n      \\le C_{d}(H_{n})^{2d-1},\n\\]\nand therefore  \n\\[\n\\mathbb{E}\\!\\Bigl[\\Bigl(\\tfrac{X_{n,d}}{(H_{n})^{d}}-1\\Bigr)^{2}\\Bigr]\n      \\le\\frac{C_{d}}{H_{n}}\n      \\xrightarrow[n\\to\\infty]{}0,\n\\]\nbecause $H_{n}\\sim\\log n\\to\\infty$.  Convergence in $L^{2}$ implies\nconvergence in probability, completing Part 3.\n\\hfill$\\square$\n\n-----------------------------------------------------------------  \nReferences  \n\n[1] G. H. Hardy and E. M. Wright, \\emph{An Introduction to the Theory of\nNumbers}, 6th ed., Oxford Univ. Press, 2008.  \n\n-----------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.419412",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimension:  the problem generalises the 1-dimensional\n   “record’’ concept to an arbitrary fixed number d ≥ 2 of dimensions,\n   increasing the state space from n to nᵈ sites.\n\n2. Additional constraints:  a site must be lighter than every crate in\n   a d-dimensional positive orthant, a much larger and overlapping set\n   of competitors than in the 1-dimensional case.\n\n3. Deeper theory:  \n   – Multi–index harmonic sums appear; separating variables is\n     essential to compute the expectation.  \n   – To bound the variance one must control covariances across a highly\n     inter-dependent family of indicator variables, requiring a careful\n     classification into comparable and incomparable pairs and an\n     estimate that hinges on geometric properties of the lattice.  \n   – Probabilistic limit results (Chebyshev together with (∗)) are\n     needed to prove convergence in probability.\n\n4. Multiple interacting concepts:  the solution blends enumerative\n   combinatorics (orthant sizes), multi–dimensional harmonic–analysis,\n   dependence estimates, and probabilistic limit theory.\n\n5. Greater length and depth:  compared with the single–sum harmonic\n   computation of the original problem, the enhanced variant demands\n   a multi–sum analysis, careful covariance bounds, and an explicit\n   asymptotic argument, each of which is several technical steps\n   longer than the entire original solution."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}