25:["$","title",null,{"children":"Discrete Calculus | Prathyush | Observable"}] 49:T568,

Normal basis

Stirling Transform

Falling factorial basis

${printPoly(parsedPoly8,true)}

${printPoly(normalToFalling(parsedPoly8), true)}

∆

${printPoly(diff(parsedPoly8), true)}

${printPoly(normalToFalling(diff(parsedPoly8)), true)}

4a:T5bc,4b:[] 29:{"id":"ee75d9609d1ed541","slug":null,"title":"Discrete Calculus","thumbnail":"2ecb9a6a9776d6f488cdb5cfeba88ef3e81be09e0419cc9c0cef11fab637040e","access_level":"public","create_time":"$D2023-06-04T21:00:49.408Z","update_time":"$D2024-08-19T16:01:33.565Z","view_time":null,"first_public_time":"$D2023-06-04T21:00:49.408Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"f33b22c086eea27f","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":"starter_2024","bio":"Software Designer","followed_by_you":false,"follows_you":false},"version":2158,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"3b0d270e1b775ddf","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":null,"bio":"Software Designer","followed_by_you":false,"follows_you":false}],"members":[{"type":"individual","id":"3b0d270e1b775ddf","login":"prathyvsh","name":"Prathyush","avatar_url":"https://avatars-next.observableusercontent.com/209a242aa240743240b9c0771686a44ba1950fce32c6180ced0d42027f5ab246","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Discrete Calculus","cells":[{"id":0,"value":"# Discrete Calculus","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":612,"value":"

What if we could represent polynomials as numbers? Here try it:

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":614,"value":"viewof poly = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":619,"value":"parsedPoly = parsePolySingleDigitString(poly);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1306,"value":"printPoly(parsedPoly, true);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1040,"value":"

What we have done here is to separate out ${poly} = ${parseInt(poly/100)} * 10² + ${parseInt(poly/10) % 10} * 10¹ + ${poly%10} * 10⁰. Then we put x in place of 10 and abstract this out into a general polynomial form as ${parseInt(poly/100)} * x² + ${parseInt(poly/10) % 10} * x¹ + ${poly%10} * x⁰.

\n \n

Conversely, we can think of a number like ${poly} as 10 → ${parseInt(poly/100)} * x² + ${parseInt(poly/10) % 10} * x¹ + ${poly%10} * x⁰. That is 10 substituted in the place of x starting from it’s polynomial representation in x.

Addition

What does an abstraction like this yield us? Turns out, we can now substitute any other number in its place. This allows us to see what this representation means in other bases other than 10.

You can do addition and subtraction in this style!

Enter the first polynomial: ${printPoly(parsePolyString(poly4))}

Enter the next polynomial:${printPoly(parsePolyString(poly5), true)}

Their sum is:

One thing you might notice is that, unlike normal number addition, there’s no carrying over after a coefficient crosses 10, but rather it stays in its position.

Multiplication

We can also multiply polynomials by a constant factor or themselves.

Enter a polynomial to multiply by:

Enter another polynomial:

Digital Root

An interesting example might be one where we replace all the xs with 1 giving us the digital root.

Now, what if we pass in an array of values into the polynomial?

1 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly), 1)}

2 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly), 2)}

3 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly), 3)}

4 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly), 4)}

Let us now see what a range of such values give us. Provide a limit and we will see what the list of values it produces.

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1136,"value":"viewof polyLimit = html``;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1227,"value":"parametrize = (poly, limit = 1) => range(limit).map((x, idx) => subst(poly, x));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1138,"value":"parametrize(parsedPoly, polyLimit).map((x, idx) => idx + \" → \" + x);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1150,"value":"

Along with 0 to a particular number, we might also want to pass in arbitrary range of numbers into our polynomial. Say, we want to find out what the values multiples of 5 gives us. This gives us the following table:

\n\n

5 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly),5)}

10 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly),10)}

15 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly),15)}

20 → ${printPoly(parsePolySingleDigitString(poly), true)} = ${subst(parsePolySingleDigitString(poly),20)}

But, now, 5, 10, 15, 20, etc. is just 5x parametrized by the natural numbers. This means that we can substitute a polynomial inside a polynomial to achieve this effect. This is called a composition operation.

This is different from multiplying two polynomials.

Multiplication:

\n${printPoly(parsedPoly)} * ${printPoly({poly: [5,0]})} = ${printPoly(multiply(parsedPoly, {poly: [5, 0]}))};\n

Composition:

\n ${printPoly(parsedPoly)} ∘ ${printPoly({poly: [5,0]})} = ${printPoly(compose(parsedPoly, {poly: [5, 0]}))};\n

What can we achieve with this? Imagine we have a series and we want to sample it at particular locations. Let’s say we have a certain polynomial:

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1329,"value":"viewof poly9 = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1355,"value":"parsedPoly9 = parsePolyString(poly9);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1333,"value":"printPoly(parsedPoly9);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1339,"value":"poly9Series = parametrize(parsePolyString(poly9), 30);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1343,"value":"

We want to see what the numbers at the square numbers are. This means, we have 1 → ${subst(parsedPoly9, 1)}, 4 → ${subst(parsedPoly9, 4)}, 9 → ${subst(parsedPoly9, 9)}, 16 → ${subst(parsedPoly9, 16)} etc. We can achieve this using a simple composition:

${printPoly(parsedPoly9, true)} ∘ ${printPoly({poly: [1, 0, 0]})} = ${printPoly(compose(parsedPoly9, {poly: [1, 0, 0]}))}

Enter the second polynomial:

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":642,"value":"viewof poly2 = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":655,"value":"{\n let p1 = parsePolySingleDigitString(poly1);\n let p2 = parsePolySingleDigitString(poly2);\n \n return html`(${printPoly(p1, true)}) + (${printPoly(p2, true)}) = ${printPoly(add(p1, p2), true)}`;\n };","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":787,"value":"

What is cool is to understand that this polynomial addition captures the special case of number addition if we substitute x = 10 in the polynomial. TODO: Implement substitution on polynomials.

Now that’s neat, but what if we want our coefficients to be greater than 9? We can achieve this by putting a comma between the literals.

Enter the first polynomial:

You might now be thinking, this is cute and all, but is there any other stuff I can do with this? Turns out you are in for a treat! There’s some really cool stuff we can do in terms of calculus on such representations.

Let us do differentiation on this polynomial

Let us confirm this by parametrizing their series by natural numbers and see if the result holds for the first few numbers:\n \n

[${range(8).map(x => subst(parsePolyString(poly8), x)).join(\",\")}]

These have as adjacent difference:

[${range(8).map(x => subst(diff(parsePolyString(poly8)), x)).join(\",\")}]

which indeed looks like the polynomial

\n${printPoly(diff(parsedPoly8))}\n

parametrized by the natural numbers

","pinned":true,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":902,"value":"rightShift = ({poly}, amount) => ({poly: poly.slice(0, -amount)});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1175,"value":"parsedPoly8 = parsePolySingleDigitString(isNaN(poly8) ? 1 : poly8);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"function diff({poly, type}) {\n\n if(type == \"falling\") {\n let unit = ({poly: [0]});\n return add(...poly.map((coeff, idx) => {\n let power = poly.length - idx - 1;\n return (power === 0 || coeff === 0) ? unit : multiply({poly: [power * coeff]}, polyFromPower(power)); }))}\n else { return fallingToNormal(diff(normalToFalling({poly}))); };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":922,"value":"polyFromPower = pow => ({poly: [1].concat(range(pow-1).map(x => 0))});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":980,"value":"diff(parsePolyString(poly8));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"sum = () => [];","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":90,"value":"range = n => [...new Array(n)].map((x, idx) => idx);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"powerToFallingPowers = (power) => ({type: \"falling\", poly: range(power + 1).map((x) => stirlingPartitions(power, power - x))})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":330,"value":"normalToFalling = ({poly}) => {\n \n let results = poly.map((coefficient, idx) => multiply({poly: [coefficient]}, powerToFallingPowers(poly.length - idx - 1)));\n\n return {poly: add(...results).poly, type: \"falling\"};;\n\n};","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":443,"value":"powerToFallingPowers(4);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":598,"value":"// printPoly({type: \"falling\", poly: [1,0,0,0,0]});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":595,"value":"// printPoly(fallingToNormal({type: \"falling\", poly: [1,0,0,0,0]}));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":352,"value":"// printPoly(fallingToNormal({poly: [3, 0, 0]}));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":356,"value":"// printPoly(normalToFalling(fallingToNormal({poly: [1, 0, 1, 0, 0]})));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":389,"value":"// printPoly(normalToFalling(fallingToNormal({poly: [1, 2, 1]})));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":502,"value":"normalToFalling({poly: [1]});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":410,"value":"// printPoly(fallingToNormal(normalToFalling({poly: [1,0,1]})));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":424,"value":"// printPoly(normalToFalling({poly: [1,1]}));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":374,"value":"// printPoly(fallingToNormal(normalToFalling({poly: [1, 0, 1, 0, 0, 1]})));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":50,"value":"combinations = (n,k) => { if(k == null) k = n; return permutations(n) / (permutations(n-k) * permutations(k)) };","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"// printPoly(powerToFallingPowers(4), true);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":644,"value":"add = (...polys) => {\n let maxLen = Math.max(...polys.map(x => x.poly.length));\n return {poly: polys.map(x => normalize(x.poly, maxLen)).reduce((x,y) => zipSum(x,y))};\n};\n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":324,"value":"// printPoly(fallingToNormal(powerToFallingPowers(1)));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":209,"value":"normalize = (x, n) => ([...new Array(Math.max(0, n - x.length))].map(x => 0)).concat(x);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":211,"value":"zipWith = op => (arr1, arr2) => (arr1.length == arr2.length) ? arr1.map((x, idx) => op(x, arr2[idx])) : \"Can’t zip\";","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":225,"value":"zipSum = zipWith((x,y) => x + y);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":231,"value":"function stirlingCycles (n, k) {\n if(n == 0 && k == 0) return 1;\n else if (n == 0 || k == 0) return 0;\n else return (n - 1) * stirlingCycles(n-1, k) + stirlingCycles(n-1, k-1);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":180,"value":"fallingPowerToPoly = (fp) => ({poly: range(fp + 1).map(x => Math.pow(-1, fp - x) * stirlingCycles(fp, x)).reverse()});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":255,"value":"fallingToNormal = ({poly}) => {\n \n let results = poly.map((coefficient, idx) => multiply({poly: [coefficient]}, fallingPowerToPoly(poly.length - idx - 1)));\n\n return add(...results);\n};","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":260,"value":"multiply = ({poly: poly1}, {poly: poly2}) => add(...poly1.map((coeff1, idx) => ({poly: leftShift({poly: poly2}, poly1.length - idx - 1).poly.map(coeff2 => (coeff1 * coeff2))})));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":841,"value":"printPoly(multiply({poly: [2,1]}, {poly: [2]}));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"sup = (idx) => String(idx).split(\"\").map(x => [\"⁰\", \"¹\",\"²\",\"³\", \"⁴\", \"⁵\", \"⁶\", \"⁷\", \"⁸\", \"⁹\"][x]);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":653,"value":"parsePolySingleDigitString = (poly) => ({poly: String(poly).split(\"\").map(x => parseInt(x))});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":40,"value":"printPoly = ({poly, type}, full = false) => {\n const term = (x, pow, format) => x === 0 ? (full ? x + \"x\" + format(pow) : \"\") : pow === 0 ? (full ? x + \"x\" + format(pow) : x) : x + \"x\" + format(pow);\n const subF = x => \"_{(\" + x + \")}\";\n const supR = x => \"^{(\" + x + \")}\";\n const sup = x => \"^{\" + x + \"}\";\n if(type == \"falling\") return html`${poly.map((x, idx) => type === \"falling\" ? term(x, poly.length - 1 - idx, subF) : term(x, poly.length - 1 - idx, supR)).filter(x => x != \"\").join(\"+\")}`;\n else return html`${poly.map((x, idx) => term(x, poly.length - 1 - idx, sup)).filter(x => x != \"\").join(\"+\")}`;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":695,"value":"parsePolyString = (str) => { \n\n try {\n let result = JSON.parse(str);\n return ({poly: result});\n } catch(e) {\n if(str.indexOf(\",]\") !== -1) return {poly: JSON.parse(str.replace(\",]\", \"]\"))};\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":18,"value":"stirlingPartitions = (n, k) => ([...new Array(k+1)].map((_, i) => (Math.pow(-1,k - i) * combinations(k, i) * Math.pow(i,n))).reduce((x, y) => x + y)) / permutations(k);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"fallingPower = (n,k) => permutations(n, k);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":822,"value":"leftShift = ({poly}, amount) => ({poly: poly.concat(range(amount).map(x => 0))})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"function permutations(n,k) { if(k == 0) { return 0 } else if(k == null) k = 1; return n <= k ? 1 : n * permutations(n - 1,k); };","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":735,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W4b","stdlib":"1"}} 24:null