{
  "index": "1993-A-3",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG"
  ],
  "difficulty": "",
  "question": "n\\}$. Let $c(n, m)$ be the number of functions $f: {\\cal P}_n \\to \\{1, 2,\n\\dots, m\\}$ such that $f(A \\cap B) = \\min\\{f(A), f(B)\\}$. Prove that\n\\[\nc(n, m) = \\sum_{j=1}^m j^n.\n\\]",
  "solution": "Solution 1. Let \\( S=\\{1,2, \\ldots, n\\} \\) and \\( S_{i}=S-\\{i\\} \\). First we show that \\( f \\mapsto\\left(f(S), f\\left(S_{1}\\right), f\\left(S_{2}\\right), \\ldots, f\\left(S_{n}\\right)\\right) \\) defines a bijection between the set of allowable functions \\( f \\) and the set of \\( (n+1) \\)-tuples \\( \\left(j, a_{1}, a_{2}, \\ldots, a_{n}\\right) \\) of elements of \\( \\{1,2, \\ldots, m\\} \\) such that \\( j \\geq a_{i} \\) for \\( i=1,2, \\ldots, n \\). This map is well defined, since for any allowable \\( f \\), we have \\( f\\left(S_{i}\\right)=f\\left(S_{i} \\cap S\\right)=\\min \\left\\{f\\left(S_{i}\\right), f(S)\\right\\} \\leq f(S) \\). The inverse map takes the tuple \\( \\left(j, a_{1}, a_{2}, \\ldots, a_{n}\\right) \\) to the function \\( f \\) such that \\( f(S)=j \\) and \\( f(T)=\\min _{i \\notin T} a_{i} \\) for \\( T \\varsubsetneqq S \\). This function satisfies \\( f(A \\cap B)=\\min \\{f(A), f(B)\\} \\), and these two constructions are inverse to each other.\n\nNow we count the \\( (n+1) \\)-tuples. For fixed \\( j \\), the possibilities for each \\( a_{i} \\) are \\( 1,2, \\ldots, j \\), so there are \\( j^{n} \\) possibilities for \\( \\left(a_{1}, a_{2}, \\ldots, a_{n}\\right) \\). Summing over \\( j \\), which can be anywhere from 1 to \\( m \\), yields \\( c(n, m)=\\sum_{j=1}^{m} j^{n} \\).\n\nSolution 2. We will use induction on \\( m \\). The base case, \\( m=1 \\), states that \\( c(n, 1)=1 \\) for all \\( n \\), which is obvious.\n\nSuppose \\( m \\geq 2 \\). Given \\( f: \\mathcal{P}_{n} \\rightarrow\\{1,2, \\ldots, m\\} \\) not identically 1 such that \\( f(A \\cap B)=\\min \\{f(A), f(B)\\} \\), let \\( S_{f} \\) be the intersection of all \\( A \\in \\mathcal{P}_{n} \\) such that \\( f(A) \\geq 2 \\). The property implies that \\( f\\left(S_{f}\\right) \\geq 2 \\), and that if \\( T \\supseteq S_{f} \\) then \\( f(T) \\geq 2 \\). Hence the sets \\( T \\) for which \\( f(T) \\geq 2 \\) are exactly those that contain \\( S_{f} \\).\n\nGiven \\( S \\in \\mathcal{P}_{n} \\), how many \\( f \\) 's have \\( S_{f}=S \\) ? To give such an \\( f \\) is the same as specifying a function\n\\[\ng:\\left\\{T \\in \\mathcal{P}_{n}: T \\supseteq S\\right\\} \\rightarrow\\{2,3, \\ldots, m\\}\n\\]\nsatisfying \\( g(A \\cap B)=\\min \\{g(A), g(B)\\} \\). There is an intersection-preserving bijection from \\( \\left\\{T \\in \\mathcal{P}_{n}: T \\supseteq S\\right\\} \\) to the set of subsets of \\( \\{1,2, \\ldots, n\\}-S \\) (the bijection maps \\( T \\) to \\( T-S \\), and an order-preserving bijection from \\( \\{2,3, \\ldots, m\\} \\) to \\( \\{1,2, \\ldots, m-1\\} \\), so the number of such functions \\( g \\) equals \\( c(n-\\# S, m-1) \\).\n\nThus, remembering to count the identically 1 function, we have\n\\[\n\\begin{aligned}\nc(n, m) & =1+\\sum_{S} c(n-\\# S, m-1) \\\\\n& =1+\\sum_{k=0}^{n}\\binom{n}{k} c(n-k, m-1) \\\\\n& =1+\\sum_{k=0}^{n}\\binom{n}{k} \\sum_{j=1}^{m-1} j^{n-k} \\quad \\text { (inductive hypothesis) } \\\\\n& =1+\\sum_{j=1}^{m-1} \\sum_{k=0}^{n}\\binom{n}{k} j^{n-k} \\\\\n& =1+\\sum_{j=1}^{m-1}(1+j)^{n} \\\\\n& =\\sum_{j=1}^{m} j^{n}\n\\end{aligned}\n\\]\ncompleting the inductive step.",
  "vars": [
    "c",
    "f",
    "A",
    "B",
    "S",
    "S_i",
    "P_n",
    "T",
    "j",
    "a_i",
    "g",
    "k",
    "S_f",
    "i"
  ],
  "params": [
    "n",
    "m"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "c": "countfunc",
        "f": "mapping",
        "A": "subseta",
        "B": "subsetb",
        "S": "wholeset",
        "S_i": "subwholeset",
        "P_n": "powersetn",
        "T": "subsett",
        "j": "indexvar",
        "a_i": "tupleelem",
        "g": "reducedmap",
        "k": "binomindex",
        "S_f": "minset",
        "i": "elemindex",
        "n": "setsize",
        "m": "rangemax"
      },
      "question": "setsize\\}$. Let $countfunc(setsize, rangemax)$ be the number of functions $mapping: {\\cal powersetn} \\to \\{1, 2,\n\\dots, rangemax\\}$ such that $mapping(subseta \\cap subsetb) = \\min\\{mapping(subseta), mapping(subsetb)\\}$. Prove that\n\\[\ncountfunc(setsize, rangemax) = \\sum_{indexvar=1}^{rangemax} indexvar^{setsize}.\n\\]",
      "solution": "Solution 1. Let \\( wholeset=\\{1,2, \\ldots, setsize\\} \\) and \\( subwholeset=wholeset-\\{elemindex\\} \\). First we show that \\( mapping \\mapsto\\left(mapping(wholeset), mapping\\left(wholeset_{1}\\right), mapping\\left(wholeset_{2}\\right), \\ldots, mapping\\left(wholeset_{setsize}\\right)\\right) \\) defines a bijection between the set of allowable functions \\( mapping \\) and the set of \\( (setsize+1) \\)-tuples \\( \\left(indexvar, tupleelem_{1}, tupleelem_{2}, \\ldots, tupleelem_{setsize}\\right) \\) of elements of \\( \\{1,2, \\ldots, rangemax\\} \\) such that \\( indexvar \\geq tupleelem_{elemindex} \\) for \\( elemindex=1,2, \\ldots, setsize \\). This map is well defined, since for any allowable \\( mapping \\), we have \\( mapping\\left(wholeset_{elemindex}\\right)=mapping\\left(wholeset_{elemindex} \\cap wholeset\\right)=\\min \\left\\{mapping\\left(wholeset_{elemindex}\\right), mapping(wholeset)\\right\\} \\leq mapping(wholeset) \\). The inverse map takes the tuple \\( \\left(indexvar, tupleelem_{1}, tupleelem_{2}, \\ldots, tupleelem_{setsize}\\right) \\) to the function \\( mapping \\) such that \\( mapping(wholeset)=indexvar \\) and \\( mapping(subsett)=\\min _{elemindex \\notin subsett} tupleelem_{elemindex} \\) for \\( subsett \\varsubsetneqq wholeset \\). This function satisfies \\( mapping(subseta \\cap subsetb)=\\min \\{mapping(subseta), mapping(subsetb)\\} \\), and these two constructions are inverse to each other.\n\nNow we count the \\( (setsize+1) \\)-tuples. For fixed \\( indexvar \\), the possibilities for each \\( tupleelem_{elemindex} \\) are \\( 1,2, \\ldots, indexvar \\), so there are \\( indexvar^{setsize} \\) possibilities for \\( \\left(tupleelem_{1}, tupleelem_{2}, \\ldots, tupleelem_{setsize}\\right) \\). Summing over \\( indexvar \\), which can be anywhere from 1 to \\( rangemax \\), yields \\( countfunc(setsize, rangemax)=\\sum_{indexvar=1}^{rangemax} indexvar^{setsize} \\).\n\nSolution 2. We will use induction on \\( rangemax \\). The base case, \\( rangemax=1 \\), states that \\( countfunc(setsize, 1)=1 \\) for all \\( setsize \\), which is obvious.\n\nSuppose \\( rangemax \\geq 2 \\). Given \\( mapping: \\mathcal{powersetn} \\rightarrow\\{1,2, \\ldots, rangemax\\} \\) not identically 1 such that \\( mapping(subseta \\cap subsetb)=\\min \\{mapping(subseta), mapping(subsetb)\\} \\), let \\( minset \\) be the intersection of all \\( subseta \\in \\mathcal{powersetn} \\) such that \\( mapping(subseta) \\geq 2 \\). The property implies that \\( mapping\\left(minset\\right) \\geq 2 \\), and that if \\( subsett \\supseteq minset \\) then \\( mapping(subsett) \\geq 2 \\). Hence the sets \\( subsett \\) for which \\( mapping(subsett) \\geq 2 \\) are exactly those that contain \\( minset \\).\n\nGiven \\( wholeset \\in \\mathcal{powersetn} \\), how many \\( mapping \\)'s have \\( minset=wholeset \\) ? To give such a \\( mapping \\) is the same as specifying a function\n\\[\nreducedmap:\\left\\{subsett \\in \\mathcal{powersetn}: subsett \\supseteq wholeset\\right\\} \\rightarrow\\{2,3, \\ldots, rangemax\\}\n\\]\nsatisfying \\( reducedmap(subseta \\cap subsetb)=\\min \\{reducedmap(subseta), reducedmap(subsetb)\\} \\). There is an intersection-preserving bijection from \\( \\left\\{subsett \\in \\mathcal{powersetn}: subsett \\supseteq wholeset\\right\\} \\) to the set of subsets of \\( \\{1,2, \\ldots, setsize\\}-wholeset \\) (the bijection maps \\( subsett \\) to \\( subsett-wholeset \\), and an order-preserving bijection from \\( \\{2,3, \\ldots, rangemax\\} \\) to \\( \\{1,2, \\ldots, rangemax-1\\} \\), so the number of such functions \\( reducedmap \\) equals \\( countfunc(setsize-\\# wholeset, rangemax-1) \\).\n\nThus, remembering to count the identically 1 function, we have\n\\[\n\\begin{aligned}\ncountfunc(setsize, rangemax) & =1+\\sum_{wholeset} countfunc(setsize-\\# wholeset, rangemax-1) \\\\\n& =1+\\sum_{binomindex=0}^{setsize}\\binom{setsize}{binomindex} countfunc(setsize-binomindex, rangemax-1) \\\\\n& =1+\\sum_{binomindex=0}^{setsize}\\binom{setsize}{binomindex} \\sum_{indexvar=1}^{rangemax-1} indexvar^{setsize-binomindex} \\quad \\text { (inductive hypothesis) } \\\\\n& =1+\\sum_{indexvar=1}^{rangemax-1} \\sum_{binomindex=0}^{setsize}\\binom{setsize}{binomindex} indexvar^{setsize-binomindex} \\\\\n& =1+\\sum_{indexvar=1}^{rangemax-1}(1+indexvar)^{setsize} \\\\\n& =\\sum_{indexvar=1}^{rangemax} indexvar^{setsize}\n\\end{aligned}\n\\]\ncompleting the inductive step."
    },
    "descriptive_long_confusing": {
      "map": {
        "c": "marigolds",
        "f": "tangerine",
        "A": "shoelaces",
        "B": "aftershave",
        "S": "toothpick",
        "S_i": "sandcastle",
        "P_n": "blueberry",
        "T": "hummingbird",
        "j": "peppercorn",
        "a_i": "butterscotch",
        "g": "chandelier",
        "k": "speedwagon",
        "S_f": "turnpike",
        "i": "raincloud",
        "n": "wilderness",
        "m": "parchment"
      },
      "question": "wilderness\\}$. Let $marigolds(wilderness, parchment)$ be the number of functions $tangerine: blueberry \\to \\{1, 2,\n\\dots, parchment\\}$ such that $tangerine(shoelaces \\cap aftershave) = \\min\\{tangerine(shoelaces), tangerine(aftershave)\\}$. Prove that\n\\[\nmarigolds(wilderness, parchment) = \\sum_{peppercorn=1}^{parchment} peppercorn^{wilderness}.\n\\]",
      "solution": "Solution 1. Let \\( toothpick=\\{1,2, \\ldots, wilderness\\} \\) and \\( sandcastle=toothpick-\\{raincloud\\} \\). First we show that \\( tangerine \\mapsto\\left(tangerine(toothpick), tangerine\\left(sandcastle_{1}\\right), tangerine\\left(sandcastle_{2}\\right), \\ldots, tangerine\\left(sandcastle_{wilderness}\\right)\\right) \\) defines a bijection between the set of allowable functions \\( tangerine \\) and the set of \\( (wilderness+1) \\)-tuples \\( \\left(peppercorn, butterscotch_{1}, butterscotch_{2}, \\ldots, butterscotch_{wilderness}\\right) \\) of elements of \\{1,2, \\ldots, parchment\\} such that \\( peppercorn \\geq butterscotch_{raincloud} \\) for \\( raincloud=1,2, \\ldots, wilderness \\). This map is well defined, since for any allowable \\( tangerine \\), we have \\(\n tangerine\\left(sandcastle_{raincloud}\\right)=tangerine\\left(sandcastle_{raincloud} \\cap toothpick\\right)=\\min \\left\\{tangerine\\left(sandcastle_{raincloud}\\right), tangerine(toothpick)\\right\\} \\leq tangerine(toothpick) .\\)\nThe inverse map takes the tuple \\( \\left(peppercorn, butterscotch_{1}, butterscotch_{2}, \\ldots, butterscotch_{wilderness}\\right) \\) to the function \\( tangerine \\) such that \\( tangerine(toothpick)=peppercorn \\) and \\( tangerine(hummingbird)=\\min _{raincloud \\notin hummingbird} butterscotch_{raincloud} \\) for \\( hummingbird \\varsubsetneqq toothpick \\). This function satisfies \\( tangerine(shoelaces \\cap aftershave)=\\min \\{tangerine(shoelaces), tangerine(aftershave)\\} \\), and these two constructions are inverse to each other.\n\nNow we count the \\( (wilderness+1) \\)-tuples. For fixed \\( peppercorn \\), the possibilities for each \\( butterscotch_{raincloud} \\) are \\( 1,2, \\ldots, peppercorn \\), so there are \\( peppercorn^{wilderness} \\) possibilities for \\( \\left(butterscotch_{1}, butterscotch_{2}, \\ldots, butterscotch_{wilderness}\\right) \\). Summing over \\( peppercorn \\), which can be anywhere from 1 to \\( parchment \\), yields \\( marigolds(wilderness, parchment)=\\sum_{peppercorn=1}^{parchment} peppercorn^{wilderness} \\).\n\nSolution 2. We will use induction on \\( parchment \\). The base case, \\( parchment=1 \\), states that \\( marigolds(wilderness, 1)=1 \\) for all \\( wilderness \\), which is obvious.\n\nSuppose \\( parchment \\geq 2 \\). Given \\( tangerine: blueberry \\rightarrow\\{1,2, \\ldots, parchment\\} \\) not identically 1 such that \\( tangerine(shoelaces \\cap aftershave)=\\min \\{tangerine(shoelaces), tangerine(aftershave)\\} \\), let \\( turnpike \\) be the intersection of all \\( shoelaces \\in blueberry \\) such that \\( tangerine(shoelaces) \\geq 2 \\). The property implies that \\( tangerine\\left(turnpike\\right) \\geq 2 \\), and that if \\( hummingbird \\supseteq turnpike \\) then \\( tangerine(hummingbird) \\geq 2 \\). Hence the sets \\( hummingbird \\) for which \\( tangerine(hummingbird) \\geq 2 \\) are exactly those that contain \\( turnpike \\).\n\nGiven \\( toothpick \\in blueberry \\), how many \\( tangerine \\)'s have \\( turnpike=toothpick \\) ? To give such a \\( tangerine \\) is the same as specifying a function\n\\[\nchandelier:\\left\\{hummingbird \\in blueberry: hummingbird \\supseteq toothpick\\right\\} \\rightarrow\\{2,3, \\ldots, parchment\\}\n\\]\nsatisfying \\( chandelier(shoelaces \\cap aftershave)=\\min \\{chandelier(shoelaces), chandelier(aftershave)\\} \\). There is an intersection-preserving bijection from \\( \\left\\{hummingbird \\in blueberry: hummingbird \\supseteq toothpick\\right\\} \\) to the set of subsets of \\( \\{1,2, \\ldots, wilderness\\}-toothpick \\) (the bijection maps \\( hummingbird \\) to \\( hummingbird-toothpick \\)), and an order-preserving bijection from \\( \\{2,3, \\ldots, parchment\\} \\) to \\( \\{1,2, \\ldots, parchment-1\\} \\), so the number of such functions \\( chandelier \\) equals \\( marigolds(wilderness-\\# toothpick, parchment-1) \\).\n\nThus, remembering to count the identically 1 function, we have\n\\[\n\\begin{aligned}\nmarigolds(wilderness, parchment) & =1+\\sum_{toothpick} marigolds(wilderness-\\# toothpick, parchment-1) \\\\\n& =1+\\sum_{speedwagon=0}^{wilderness}\\binom{wilderness}{speedwagon} marigolds(wilderness-speedwagon, parchment-1) \\\\\n& =1+\\sum_{speedwagon=0}^{wilderness}\\binom{wilderness}{speedwagon} \\sum_{peppercorn=1}^{parchment-1} peppercorn^{wilderness-speedwagon} \\quad \\text { (inductive hypothesis) } \\\\\n& =1+\\sum_{peppercorn=1}^{parchment-1} \\sum_{speedwagon=0}^{wilderness}\\binom{wilderness}{speedwagon} peppercorn^{wilderness-speedwagon} \\\\\n& =1+\\sum_{peppercorn=1}^{parchment-1}(1+peppercorn)^{wilderness} \\\\\n& =\\sum_{peppercorn=1}^{parchment} peppercorn^{wilderness}\n\\end{aligned}\n\\]\ncompleting the inductive step."
    },
    "descriptive_long_misleading": {
      "map": {
        "c": "scarcity",
        "f": "dysfunction",
        "A": "universalset",
        "B": "emptyworld",
        "S": "emptysource",
        "S_i": "emptysourceprime",
        "P_n": "singleton",
        "T": "outsiderset",
        "j": "fixedpoint",
        "a_i": "flexiblevalue",
        "g": "malfunction",
        "k": "infinite",
        "S_f": "targetfset",
        "i": "aggregate",
        "n": "nothingness",
        "m": "voidness"
      },
      "question": "nothingness\\}$. Let $scarcity(nothingness, voidness)$ be the number of functions $dysfunction: {\\cal singleton}_{nothingness} \\to \\{1, 2, \\dots, voidness\\}$ such that $dysfunction(universalset \\cap emptyworld) = \\min\\{dysfunction(universalset), dysfunction(emptyworld)\\}$. Prove that\\n\\[\\nscarcity(nothingness, voidness) = \\sum_{fixedpoint=1}^{voidness} fixedpoint^{nothingness}.\\n\\]",
      "solution": "Solution 1. Let \\( emptysource=\\{1,2, \\ldots, nothingness\\} \\) and \\( emptysourceprime=emptysource-\\{aggregate\\} \\). First we show that \\( dysfunction \\mapsto\\left(dysfunction(emptysource), dysfunction\\left(emptysourceprime\\right), dysfunction\\left(emptysourceprime\\right), \\ldots, dysfunction\\left(emptysourceprime\\right)\\right) \\) defines a bijection between the set of allowable functions \\( dysfunction \\) and the set of \\( (nothingness+1) \\)-tuples \\( \\left(fixedpoint, flexiblevalue_{1}, flexiblevalue_{2}, \\ldots, flexiblevalue_{nothingness}\\right) \\) of elements of \\( \\{1,2, \\ldots, voidness\\} \\) such that \\( fixedpoint \\geq flexiblevalue_{\\aggregate} \\) for \\( \\aggregate=1,2, \\ldots, nothingness \\). This map is well defined, since for any allowable \\( dysfunction \\), we have \\( dysfunction\\left(emptysourceprime\\right)=dysfunction\\left(emptysourceprime \\cap emptysource\\right)=\\min \\left\\{dysfunction\\left(emptysourceprime\\right), dysfunction(emptysource)\\right\\} \\leq dysfunction(emptysource) \\). The inverse map takes the tuple \\( \\left(fixedpoint, flexiblevalue_{1}, flexiblevalue_{2}, \\ldots, flexiblevalue_{nothingness}\\right) \\) to the function \\( dysfunction \\) such that \\( dysfunction(emptysource)=fixedpoint \\) and \\( dysfunction(outsiderset)=\\min _{\\aggregate \\notin outsiderset} flexiblevalue_{\\aggregate} \\) for \\( outsiderset \\varsubsetneqq emptysource \\). This function satisfies \\( dysfunction(universalset \\cap emptyworld)=\\min \\{dysfunction(universalset), dysfunction(emptyworld)\\} \\), and these two constructions are inverse to each other.\\n\\nNow we count the \\( (nothingness+1) \\)-tuples. For fixed \\( fixedpoint \\), the possibilities for each \\( flexiblevalue_{\\aggregate} \\) are \\( 1,2, \\ldots, fixedpoint \\), so there are \\( fixedpoint^{nothingness} \\) possibilities for \\( \\left(flexiblevalue_{1}, flexiblevalue_{2}, \\ldots, flexiblevalue_{nothingness}\\right) \\). Summing over \\( fixedpoint \\), which can be anywhere from 1 to \\( voidness \\), yields \\( scarcity(nothingness, voidness)=\\sum_{fixedpoint=1}^{voidness} fixedpoint^{nothingness} \\).\\n\\nSolution 2. We will use induction on \\( voidness \\). The base case, \\( voidness=1 \\), states that \\( scarcity(nothingness, 1)=1 \\) for all \\( nothingness \\), which is obvious.\\n\\nSuppose \\( voidness \\geq 2 \\). Given \\( dysfunction: \\mathcal{singleton}_{nothingness} \\rightarrow\\{1,2, \\ldots, voidness\\} \\) not identically 1 such that \\( dysfunction(universalset \\cap emptyworld)=\\min \\{dysfunction(universalset), dysfunction(emptyworld)\\} \\), let \\( targetfset \\) be the intersection of all \\( universalset \\in \\mathcal{singleton}_{nothingness} \\) such that \\( dysfunction(universalset) \\geq 2 \\). The property implies that \\( dysfunction\\left(targetfset\\right) \\geq 2 \\), and that if \\( outsiderset \\supseteq targetfset \\) then \\( dysfunction(outsiderset) \\geq 2 \\). Hence the sets \\( outsiderset \\) for which \\( dysfunction(outsiderset) \\geq 2 \\) are exactly those that contain \\( targetfset \\).\\n\\nGiven \\( emptysource \\in \\mathcal{singleton}_{nothingness} \\), how many \\( dysfunction \\)'s have \\( targetfset=emptysource \\) ? To give such an \\( dysfunction \\) is the same as specifying a function\\n\\[\\nmalfunction:\\left\\{outsiderset \\in \\mathcal{singleton}_{nothingness}: outsiderset \\supseteq emptysource\\right\\} \\rightarrow\\{2,3, \\ldots, voidness\\}\\n\\]satisfying \\( malfunction(universalset \\cap emptyworld)=\\min \\{malfunction(universalset), malfunction(emptyworld)\\} \\). There is an intersection-preserving bijection from \\( \\left\\{outsiderset \\in \\mathcal{singleton}_{nothingness}: outsiderset \\supseteq emptysource\\right\\} \\) to the set of subsets of \\( \\{1,2, \\ldots, nothingness\\}-emptysource \\) (the bijection maps \\( outsiderset \\) to \\( outsiderset-emptysource \\)), and an order-preserving bijection from \\( \\{2,3, \\ldots, voidness\\} \\) to \\( \\{1,2, \\ldots, voidness-1\\} \\), so the number of such functions \\( malfunction \\) equals \\( scarcity(nothingness-\\# emptysource, voidness-1) \\).\\n\\nThus, remembering to count the identically 1 function, we have\\n\\[\\n\\begin{aligned}\\nscarcity(nothingness, voidness) & =1+\\sum_{emptysource} scarcity(nothingness-\\# emptysource, voidness-1) \\\\n& =1+\\sum_{infinite=0}^{nothingness}\\binom{nothingness}{infinite} scarcity(nothingness-infinite, voidness-1) \\\\\n& =1+\\sum_{infinite=0}^{nothingness}\\binom{nothingness}{infinite} \\sum_{fixedpoint=1}^{voidness-1} fixedpoint^{nothingness-infinite} \\quad \\text { (inductive hypothesis) } \\\\\n& =1+\\sum_{fixedpoint=1}^{voidness-1} \\sum_{infinite=0}^{nothingness}\\binom{nothingness}{infinite} fixedpoint^{nothingness-infinite} \\\\\n& =1+\\sum_{fixedpoint=1}^{voidness-1}(1+fixedpoint)^{nothingness} \\\\\n& =\\sum_{fixedpoint=1}^{voidness} fixedpoint^{nothingness}\\n\\end{aligned}\\n\\]completing the inductive step."
    },
    "garbled_string": {
      "map": {
        "c": "qzxwvtnp",
        "f": "hjgrksla",
        "A": "odfjaklq",
        "B": "mveursta",
        "S": "plinwqer",
        "S_i": "bclamnso",
        "P_n": "truvynxe",
        "T": "spqodkwe",
        "j": "fguiolek",
        "a_i": "lkjwerpo",
        "g": "vxzeqmur",
        "k": "dfghjklo",
        "S_f": "qrmnplka",
        "i": "asdfghjk",
        "n": "poijhuyt",
        "m": "wqazlkjh"
      },
      "question": "poijhuyt\\}$. Let $qzxwvtnp(poijhuyt, wqazlkjh)$ be the number of functions $hjgrksla: {\\cal truvynxe}_{poijhuyt} \\to \\{1, 2, \\dots, wqazlkjh\\}$ such that $hjgrksla(odfjaklq \\cap mveursta) = \\min\\{hjgrksla(odfjaklq), hjgrksla(mveursta)\\}$. Prove that\n\\[\nqzxwvtnp(poijhuyt, wqazlkjh) = \\sum_{fguiolek=1}^{wqazlkjh} fguiolek^{poijhuyt}.\n\\]",
      "solution": "Solution 1. Let \\( plinwqer=\\{1,2, \\ldots, poijhuyt\\} \\) and \\( bclamnso=plinwqer-\\{asdfghjk\\} \\). First we show that \\( hjgrksla \\mapsto\\left(hjgrksla(plinwqer), hjgrksla\\left(bclamnso\\right), hjgrksla\\left(bclamnso\\right), \\ldots, hjgrksla\\left(bclamnso\\right)\\right) \\) defines a bijection between the set of allowable functions \\( hjgrksla \\) and the set of \\( (poijhuyt+1) \\)-tuples \\( \\left(fguiolek, lkjwerpo, lkjwerpo, \\ldots, lkjwerpo\\right) \\) of elements of \\( \\{1,2, \\ldots, wqazlkjh\\} \\) such that \\( fguiolek \\geq lkjwerpo \\) for \\( asdfghjk=1,2, \\ldots, poijhuyt \\). This map is well defined, since for any allowable \\( hjgrksla \\), we have \\( hjgrksla\\left(bclamnso\\right)=hjgrksla\\left(bclamnso \\cap plinwqer\\right)=\\min \\left\\{hjgrksla\\left(bclamnso\\right), hjgrksla(plinwqer)\\right\\} \\leq hjgrksla(plinwqer) \\). The inverse map takes the tuple \\( \\left(fguiolek, lkjwerpo, lkjwerpo, \\ldots, lkjwerpo\\right) \\) to the function \\( hjgrksla \\) such that \\( hjgrksla(plinwqer)=fguiolek \\) and \\( hjgrksla(spqodkwe)=\\min _{asdfghjk \\notin spqodkwe} lkjwerpo \\) for \\( spqodkwe \\varsubsetneqq plinwqer \\). This function satisfies \\( hjgrksla(odfjaklq \\cap mveursta)=\\min \\{hjgrksla(odfjaklq), hjgrksla(mveursta)\\} \\), and these two constructions are inverse to each other.\n\nNow we count the \\( (poijhuyt+1) \\)-tuples. For fixed \\( fguiolek \\), the possibilities for each \\( lkjwerpo \\) are \\( 1,2, \\ldots, fguiolek \\), so there are \\( fguiolek^{poijhuyt} \\) possibilities for \\( \\left(lkjwerpo, lkjwerpo, \\ldots, lkjwerpo\\right) \\). Summing over \\( fguiolek \\), which can be anywhere from 1 to \\( wqazlkjh \\), yields \\( qzxwvtnp(poijhuyt, wqazlkjh)=\\sum_{fguiolek=1}^{wqazlkjh} fguiolek^{poijhuyt} \\).\n\nSolution 2. We will use induction on \\( wqazlkjh \\). The base case, \\( wqazlkjh=1 \\), states that \\( qzxwvtnp(poijhuyt, 1)=1 \\) for all \\( poijhuyt \\), which is obvious.\n\nSuppose \\( wqazlkjh \\geq 2 \\). Given \\( hjgrksla: truvynxe_{poijhuyt} \\rightarrow\\{1,2, \\ldots, wqazlkjh\\} \\) not identically 1 such that \\( hjgrksla(odfjaklq \\cap mveursta)=\\min \\{hjgrksla(odfjaklq), hjgrksla(mveursta)\\} \\), let \\( qrmnplka \\) be the intersection of all \\( odfjaklq \\in truvynxe_{poijhuyt} \\) such that \\( hjgrksla(odfjaklq) \\geq 2 \\). The property implies that \\( hjgrksla\\left(qrmnplka\\right) \\geq 2 \\), and that if \\( spqodkwe \\supseteq qrmnplka \\) then \\( hjgrksla(spqodkwe) \\geq 2 \\). Hence the sets \\( spqodkwe \\) for which \\( hjgrksla(spqodkwe) \\geq 2 \\) are exactly those that contain \\( qrmnplka \\).\n\nGiven \\( plinwqer \\in truvynxe_{poijhuyt} \\), how many \\( hjgrksla \\)'s have \\( qrmnplka=plinwqer \\) ? To give such an \\( hjgrksla \\) is the same as specifying a function\n\\[\nvxzeqmur:\\left\\{spqodkwe \\in truvynxe_{poijhuyt}: spqodkwe \\supseteq plinwqer\\right\\} \\rightarrow\\{2,3, \\ldots, wqazlkjh\\}\n\\]\nsatisfying \\( vxzeqmur(odfjaklq \\cap mveursta)=\\min \\{vxzeqmur(odfjaklq), vxzeqmur(mveursta)\\} \\). There is an intersection-preserving bijection from \\( \\left\\{spqodkwe \\in truvynxe_{poijhuyt}: spqodkwe \\supseteq plinwqer\\right\\} \\) to the set of subsets of \\( \\{1,2, \\ldots, poijhuyt\\}-plinwqer \\) (the bijection maps \\( spqodkwe \\) to \\( spqodkwe-plinwqer \\)), and an order-preserving bijection from \\( \\{2,3, \\ldots, wqazlkjh\\} \\) to \\( \\{1,2, \\ldots, wqazlkjh-1\\} \\), so the number of such functions \\( vxzeqmur \\) equals \\( qzxwvtnp(poijhuyt-\\# plinwqer, wqazlkjh-1) \\).\n\nThus, remembering to count the identically 1 function, we have\n\\[\n\\begin{aligned}\nqzxwvtnp(poijhuyt, wqazlkjh) & =1+\\sum_{plinwqer} qzxwvtnp(poijhuyt-\\# plinwqer, wqazlkjh-1) \\\\\n& =1+\\sum_{dfghjklo=0}^{poijhuyt}\\binom{poijhuyt}{dfghjklo} qzxwvtnp(poijhuyt-dfghjklo, wqazlkjh-1) \\\\\n& =1+\\sum_{dfghjklo=0}^{poijhuyt}\\binom{poijhuyt}{dfghjklo} \\sum_{fguiolek=1}^{wqazlkjh-1} fguiolek^{poijhuyt-dfghjklo} \\quad \\text { (inductive hypothesis) } \\\\\n& =1+\\sum_{fguiolek=1}^{wqazlkjh-1} \\sum_{dfghjklo=0}^{poijhuyt}\\binom{poijhuyt}{dfghjklo} fguiolek^{poijhuyt-dfghjklo} \\\\\n& =1+\\sum_{fguiolek=1}^{wqazlkjh-1}(1+fguiolek)^{poijhuyt} \\\\\n& =\\sum_{fguiolek=1}^{wqazlkjh} fguiolek^{poijhuyt}\n\\end{aligned}\n\\]\ncompleting the inductive step."
    },
    "kernel_variant": {
      "question": "Let r, k, d be positive integers with k ≥ r and d ≥ 2, and put  \n  X = {1,2,…,r}.  \n\nWrite Lₖ = {0,1,…,k−1} equipped with the usual order; on the direct product  \nLₖᵈ we use the product order and denote the coordinate–wise join and meet by  \n∨ and ∧.  \n\nA map  \n  h : 𝒫(X) → Lₖᵈ  \nis called a synchronised d-lattice embedding if it fulfils  \n\n(J)  h(A ∪ B) = h(A) ∨ h(B) for all A,B ⊆ X;             (join preservation)  \n\n(I)  the r vectors h({1}),…,h({r}) are pairwise distinct;    (injectivity on atoms)  \n\n(S) there exists a permutation π of X such that for every coordinate  \n  t = 1,…,d we have  \n   h_t({π(1)}) < h_t({π(2)}) < … < h_t({π(r)}).           (common strict order)  \n\nLet D_d(r,k) be the number of such maps.  \nProve that  \n\n                      D_d(r,k) = r! · ( k choose r )ᵈ. (★)",
      "solution": "Step 1 – A normal form for synchronised embeddings.  \nFor any h satisfying (J) its value on the empty set is forced:\n\n  h(∅) = h(∅ ∪ ∅) = h(∅) ∨ h(∅) = h(∅),\n\nwhile (J) with B = ∅ also gives h(A) ≥ h(∅).  \nHence put 0⃗ := (0,…,0).  \nBecause 0⃗ is the unique minimal element of Lₖᵈ, condition (J) compels\n\n  h(∅) = 0⃗.                                                  (1)\n\nMoreover, writing v_i := h({i}) (1 ≤ i ≤ r), repeated use of (J) yields\n\n  h(S) = ⋁_{i∈S} v_i  for every non-empty S ⊆ X.     (2)\n\nIndeed, if S = {i₁,…,i_t}, iterate (J):\n\n h({i₁} ∪ … ∪ {i_t}) = v_{i₁} ∨ … ∨ v_{i_t}.\n\nThus once the atom–values (v_1,…,v_r) are known, the whole map h is fixed by\n(1)–(2).  The essential task is therefore to determine which r-tuples\n(v_1,…,v_r) ⊂ Lₖᵈ give rise to a synchronised embedding.\n\nStep 2 – Characterising admissible atom-images.  \nCondition (I) merely requires v_i ≠ v_j for i ≠ j.  \nCondition (S) is the real obstruction: there must exist one permutation π with\n\n  v_{π(1)}(t) < v_{π(2)}(t) < … < v_{π(r)}(t) for every t = 1,…,d, (3)\n\nwhere v_j(t) denotes the t-th coordinate of v_j.  In words, the SAME ordering\nof the indices must simultaneously sort the singletons strictly increasingly\nin every coordinate.\n\nLemma.  A family (v_1,…,v_r) ⊂ Lₖᵈ satisfies (3) iff\n\n (a) for every coordinate t the r numbers {v_1(t),…,v_r(t)} are distinct; and  \n (b) for any indices i,j we have  \n  v_i(t) < v_j(t) ⇔ v_i(s) < v_j(s) for all 1 ≤ t,s ≤ d.\n\nProof.  (3) clearly implies (a) and (b).  Conversely, if (a) and (b) hold,\nlet σ be the permutation which orders the first coordinate increasingly;\nthen (b) forces σ to order EVERY coordinate increasingly, yielding (3). ∎\n\nCondition (b) says that all d coordinate lists are *coherently ordered*;\nhence to construct (v_1,…,v_r) we may proceed as follows:\n\n • Choose any permutation π of {1,…,r}.  \n • In coordinate 1 choose an r-element *increasing* subset of Lₖ,\n  say 0 ≤ α_1 < α_2 < … < α_r ≤ k−1; there are (k choose r) ways.  \n • For each remaining coordinate t = 2,…,d choose another increasing\n  r-tuple 0 ≤ β_1^{(t)} < … < β_r^{(t)} ≤ k−1, again in (k choose r) ways.  \n • Finally put  \n  v_{π(j)}(1) = α_j and v_{π(j)}(t) = β_j^{(t)}  (2 ≤ t ≤ d).\n\nBecause the same index j is used in every coordinate, property (3) is met.\nConversely, any admissible family arises uniquely in this fashion.\n\nHence the number of admissible (v_1,…,v_r) equals\n\n  r! · ( k choose r )ᵈ.                                      (4)\n\nStep 3 – Extending atom-values to a lattice embedding.  \nTake an admissible (v_1,…,v_r).  Define h by (1)–(2).  \nWe must check that h indeed satisfies (J).\n\nLet A,B ⊆ X.  Using (2),\n\n h(A) ∨ h(B) = (⋁_{i∈A} v_i) ∨ (⋁_{j∈B} v_j)\n       = ⋁_{i∈A∪B} v_i = h(A ∪ B),\n\nbecause join is associative, commutative, and idempotent in Lₖᵈ.  \nThus (J) holds.  Conditions (I) and (S) were built in, so h is a synchronised\nd-lattice embedding.\n\nConversely, any synchronised embedding determines its atom-values, which are\nadmissible in the sense of Step 2.  Consequently, the construction is\nbijective.\n\nStep 4 – Counting embeddings.  \nBy (4) each admissible atom-tuple – equivalently each synchronised embedding –\nis counted exactly once, proving\n\n  D_d(r,k) = r! · ( k choose r )ᵈ.  □",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.730392",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original kernel variant, the enhanced problem raises the bar\nin several independent ways.\n\n1. Higher-dimensional codomain:  Instead of a simple chain {0,…,k−1}, we work\n   in the d-fold product Lₖᵈ, a distributive lattice of dimension d ≥ 2.\n   Join now acts *coordinate-wise*, which forces the solver to reason about\n   vectors rather than scalars.\n\n2. Synchronised ordering constraint (S):  All coordinates must be ordered by\n   the same permutation.  This couples the d components in a non-trivial way\n   and excludes any coordinate-wise decomposition of the problem.\n\n3. Lattice-theoretic viewpoint:  One must recognise that (J) turns the problem\n   into counting lattice homomorphisms from the Boolean algebra 𝒫(X) to\n   Lₖᵈ, and that such homomorphisms are completely determined by their values\n   on atoms.  This step, absent from the original problem, requires\n   background in lattice theory.\n\n4. Injection vs. coherence:  In the original kernel variant the singletons\n   only had to be pairwise different.  Here they must be *simultaneously*\n   strictly ordered in every coordinate, introducing a delicate compatibility\n   condition (Lemma in Step 2).\n\n5. Enumeration complexity:  The final count involves a permutation factor and\n   d independent but *synchronised* selections of r-subsets of size k,\n   leading to the product r!·(k choose r)ᵈ.  Deriving this count demands\n   Möbius-style reasoning about admissibility and uniqueness, rather than the\n   straightforward falling-factorial sum Σ_{s=r}^k s!/(s−r)! of the kernel\n   variant.\n\nBecause of these additional structural, dimensional, and combinatorial\ndemands, solving the enhanced variant requires deeper insight and a broader\ntool-set than either the original problem or the current kernel version."
      }
    },
    "original_kernel_variant": {
      "question": "Let r, k, d be positive integers with k ≥ r and d ≥ 2, and put  \n  X = {1,2,…,r}.  \n\nWrite Lₖ = {0,1,…,k−1} equipped with the usual order; on the direct product  \nLₖᵈ we use the product order and denote the coordinate–wise join and meet by  \n∨ and ∧.  \n\nA map  \n  h : 𝒫(X) → Lₖᵈ  \nis called a synchronised d-lattice embedding if it fulfils  \n\n(J)  h(A ∪ B) = h(A) ∨ h(B) for all A,B ⊆ X;             (join preservation)  \n\n(I)  the r vectors h({1}),…,h({r}) are pairwise distinct;    (injectivity on atoms)  \n\n(S) there exists a permutation π of X such that for every coordinate  \n  t = 1,…,d we have  \n   h_t({π(1)}) < h_t({π(2)}) < … < h_t({π(r)}).           (common strict order)  \n\nLet D_d(r,k) be the number of such maps.  \nProve that  \n\n                      D_d(r,k) = r! · ( k choose r )ᵈ. (★)",
      "solution": "Step 1 – A normal form for synchronised embeddings.  \nFor any h satisfying (J) its value on the empty set is forced:\n\n  h(∅) = h(∅ ∪ ∅) = h(∅) ∨ h(∅) = h(∅),\n\nwhile (J) with B = ∅ also gives h(A) ≥ h(∅).  \nHence put 0⃗ := (0,…,0).  \nBecause 0⃗ is the unique minimal element of Lₖᵈ, condition (J) compels\n\n  h(∅) = 0⃗.                                                  (1)\n\nMoreover, writing v_i := h({i}) (1 ≤ i ≤ r), repeated use of (J) yields\n\n  h(S) = ⋁_{i∈S} v_i  for every non-empty S ⊆ X.     (2)\n\nIndeed, if S = {i₁,…,i_t}, iterate (J):\n\n h({i₁} ∪ … ∪ {i_t}) = v_{i₁} ∨ … ∨ v_{i_t}.\n\nThus once the atom–values (v_1,…,v_r) are known, the whole map h is fixed by\n(1)–(2).  The essential task is therefore to determine which r-tuples\n(v_1,…,v_r) ⊂ Lₖᵈ give rise to a synchronised embedding.\n\nStep 2 – Characterising admissible atom-images.  \nCondition (I) merely requires v_i ≠ v_j for i ≠ j.  \nCondition (S) is the real obstruction: there must exist one permutation π with\n\n  v_{π(1)}(t) < v_{π(2)}(t) < … < v_{π(r)}(t) for every t = 1,…,d, (3)\n\nwhere v_j(t) denotes the t-th coordinate of v_j.  In words, the SAME ordering\nof the indices must simultaneously sort the singletons strictly increasingly\nin every coordinate.\n\nLemma.  A family (v_1,…,v_r) ⊂ Lₖᵈ satisfies (3) iff\n\n (a) for every coordinate t the r numbers {v_1(t),…,v_r(t)} are distinct; and  \n (b) for any indices i,j we have  \n  v_i(t) < v_j(t) ⇔ v_i(s) < v_j(s) for all 1 ≤ t,s ≤ d.\n\nProof.  (3) clearly implies (a) and (b).  Conversely, if (a) and (b) hold,\nlet σ be the permutation which orders the first coordinate increasingly;\nthen (b) forces σ to order EVERY coordinate increasingly, yielding (3). ∎\n\nCondition (b) says that all d coordinate lists are *coherently ordered*;\nhence to construct (v_1,…,v_r) we may proceed as follows:\n\n • Choose any permutation π of {1,…,r}.  \n • In coordinate 1 choose an r-element *increasing* subset of Lₖ,\n  say 0 ≤ α_1 < α_2 < … < α_r ≤ k−1; there are (k choose r) ways.  \n • For each remaining coordinate t = 2,…,d choose another increasing\n  r-tuple 0 ≤ β_1^{(t)} < … < β_r^{(t)} ≤ k−1, again in (k choose r) ways.  \n • Finally put  \n  v_{π(j)}(1) = α_j and v_{π(j)}(t) = β_j^{(t)}  (2 ≤ t ≤ d).\n\nBecause the same index j is used in every coordinate, property (3) is met.\nConversely, any admissible family arises uniquely in this fashion.\n\nHence the number of admissible (v_1,…,v_r) equals\n\n  r! · ( k choose r )ᵈ.                                      (4)\n\nStep 3 – Extending atom-values to a lattice embedding.  \nTake an admissible (v_1,…,v_r).  Define h by (1)–(2).  \nWe must check that h indeed satisfies (J).\n\nLet A,B ⊆ X.  Using (2),\n\n h(A) ∨ h(B) = (⋁_{i∈A} v_i) ∨ (⋁_{j∈B} v_j)\n       = ⋁_{i∈A∪B} v_i = h(A ∪ B),\n\nbecause join is associative, commutative, and idempotent in Lₖᵈ.  \nThus (J) holds.  Conditions (I) and (S) were built in, so h is a synchronised\nd-lattice embedding.\n\nConversely, any synchronised embedding determines its atom-values, which are\nadmissible in the sense of Step 2.  Consequently, the construction is\nbijective.\n\nStep 4 – Counting embeddings.  \nBy (4) each admissible atom-tuple – equivalently each synchronised embedding –\nis counted exactly once, proving\n\n  D_d(r,k) = r! · ( k choose r )ᵈ.  □",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.567378",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original kernel variant, the enhanced problem raises the bar\nin several independent ways.\n\n1. Higher-dimensional codomain:  Instead of a simple chain {0,…,k−1}, we work\n   in the d-fold product Lₖᵈ, a distributive lattice of dimension d ≥ 2.\n   Join now acts *coordinate-wise*, which forces the solver to reason about\n   vectors rather than scalars.\n\n2. Synchronised ordering constraint (S):  All coordinates must be ordered by\n   the same permutation.  This couples the d components in a non-trivial way\n   and excludes any coordinate-wise decomposition of the problem.\n\n3. Lattice-theoretic viewpoint:  One must recognise that (J) turns the problem\n   into counting lattice homomorphisms from the Boolean algebra 𝒫(X) to\n   Lₖᵈ, and that such homomorphisms are completely determined by their values\n   on atoms.  This step, absent from the original problem, requires\n   background in lattice theory.\n\n4. Injection vs. coherence:  In the original kernel variant the singletons\n   only had to be pairwise different.  Here they must be *simultaneously*\n   strictly ordered in every coordinate, introducing a delicate compatibility\n   condition (Lemma in Step 2).\n\n5. Enumeration complexity:  The final count involves a permutation factor and\n   d independent but *synchronised* selections of r-subsets of size k,\n   leading to the product r!·(k choose r)ᵈ.  Deriving this count demands\n   Möbius-style reasoning about admissibility and uniqueness, rather than the\n   straightforward falling-factorial sum Σ_{s=r}^k s!/(s−r)! of the kernel\n   variant.\n\nBecause of these additional structural, dimensional, and combinatorial\ndemands, solving the enhanced variant requires deeper insight and a broader\ntool-set than either the original problem or the current kernel version."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}