catalan

table.tsv

@@ -659,6 +659,7 @@ Iv⊢⍤⍀Y	Expand leading axis of Y (forces ⍀ to be a function even with a f
 ⍉∘⍪Yv	Forming first row of a matrix for later expansion	Tacit	Monadic Function		Structural	default initialise initialize 1st table	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQe9W5RAAFDBSMFYy4kwUe9nY86ZjzqXQWR@v8fAA	
 ⍉∘⍪Yv	Reshaping vector Yv into a one-row matrix	Tacit	Monadic Function		Structural	1-row table list	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQe9W5RAAFDBSMFYy4kwUe9nY86ZjzqXQWR@v8fAA	
 M+.×N	Dot/Vector/Cross/Matrix Product of M and N (¯1↑⍴M ↔ 1↑⍴N)	Tacit	Dyadic Function		Mathematical	dotproduct vectorproduct crossproduct ∙ ⋅ · multiplication multiply		
+A+.×Nv	Summation over subsets of Nv specified by rows of A	Tacit	Dyadic Function		Mathematical		https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQ0TBSMH/VuMVQwUDDUAWIgaQglDRQMNBW09Q5PVzAGipj8/w8A	
 M-.×N	Alternating Matrix Product of M and N (¯1↑⍴M ↔ 1↑⍴N)	Tacit	Dyadic Function		Mathematical			
 Am∨.∧Bm	Extending a transitive binary relation	Tacit	Dyadic Function		Boolean/Logical	extension base-2 base2		
 Mv⌈.×Nv	Maximum of Nv with weights Mv	Tacit	Dyadic Function		Mathematical	greatestif largest biggest		
@@ -711,7 +712,6 @@ I∊⍨∘⍳Jv	Boolean array of shape Jv with ones in locations Iv (inverse of
 ,[⍳2]Y	Merge the leading two axes of Y	Tacit	Monadic Function		Structural	laminate ravel combine dimensions melt fuse join along leading axis	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQSFUCkkYKxgsmj3i2PejcbmXBBpYB8hUQYRycaJBeLEADJwgX//wcA	
 ⊢∘≢⌸Y	Count of occurrences of each unique major cell	Tacit	Monadic Function		Mathematical	elements items how many repetitions times repeated		
 +.×⍨Nv	Sum of squares of Nv	Tacit	Dyadic Function		Mathematical	∑x² Σx² sum(x^2) Pythagorean trigonometry		
-A+.×Nv	Summation over subsets of Nv specified by rows of A	Tacit	Dyadic Function		Mathematical		https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQ0TBSMH/VuMVQwUDDUAWIgaQglDRQMNBW09Q5PVzAGipj8/w8A	
 Xs∨.≠⍨Y	Do rows of Y contain elements differing from Xs?	Tacit	Dyadic Function		Comparison	testif hasdifferent cells items		
 ∘.⊢⍨Yv	Square matrix with Yv as rows	Tacit	Monadic Function		Structural	tomatrixof table	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQedczQe9S16FHvCgVDBSMFYwWT//8B	
 ∘.⊣⍨Yv	Square matrix with Yv as columns	Tacit	Monadic Function		Structural	tomatrixof table	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExQedczQe9S1@FHvCgVDBSMFYwWT//8B	
@@ -2041,6 +2041,7 @@ Av{⍵⌷⍨⊂z⍳⌈\z←⍋g[⍋(+\⍺)[g←⍒⍵]]}Y	Cumulative minimum (
 Iv{⍺(≥∘(2⊃⊢,-)∧≤∘⊃)⍨-/¨⍳2⍴⍵}Jv	Iv (k or k₁,k₂) band matrix of size Jv (n or m,n)	Dfn	Dyadic Function		Mathematical	banded tridiagonal pentadiagonal heptadiagonal tri-diagonal penta-diagonal hepta-diagonal	https://tio.run/##AVMArP9hcGwtZHlhbG9n///ijpXihpAxIMKvMiB74o26KOKJpeKImCgy4oqD4oqiLC0p4oin4omk4oiY4oqDKeKNqC0vwqjijbMy4o204o21fSA1IDb//w	
 {⍵ ⍵⍴⍋+\a/(≢a←1↓⌽2/⍳⍵)⍴,∘-⍨1⍵}Js	Number Spiral of order Js	Dfn	Monadic Function		Mathematical	rotation concentric rectangular rings eVolutes spirals & time	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExSqH/VuVQDiR71bHvV2a8ck6ms86lyUCJQyfNQ2@VHPXiP9R72bgQo0gSp0HnXM0H3Uu8IQyK9VMP3/HwA	
 Iv{⍵⍴⍨⍺×@(<∘0)⍨(×/⍴⍵)÷×/⍺~0}Y	Reshape (⍴) Y to shape Iv, allowing ¯1 to automatically determine missing length	Dfn	Dyadic Function		Structural	mold form dimension array autoshape axis		
+{⌽⊃¨{⍵,⍨⊂1↓+\⊃⍵}⍣(⍵-1)⊂⍵↑1}Js	First Js Catalan numbers (number of ways to nest Js pairs of parentheses)	Dfn	Monadic Function		Mathematical	numbers dyck words binary trees mountain ranges	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExSqH/XsfdTVfGhF9aPerTqPelc86moyfNQ2WTsGKAoUqn3Uu1gDSOsaagJlgIxHbRMNaxUMDf7/BwA	
 Iv{(⍳(≢⍵)++/⍺)∊+\1+¯1↓0,⍺\⍨1⌽⍵}Bv	Expansion vector (left argument for \ or ⍀) to insert Iv[i] elements after i'th subvector (subvectors indicated by Bv)	Dfn	Dyadic Function		Structural	expand openup major cells rows items lists		
 {⍺←⎕UCS 10 ⋄ 1↓(,1,⌽∨\⌽' '≠⍵)/,⍺,⍵}Cm	Linefeed-separated character vector constructed from the rows of Dm (which must be of depth 1) with trailing spaces removed	Dfn	Monadic Function		Structural	text dtb blanks lists ending split from table matrix to removing delimited segmented LF newline string segmented MATtoSS separated separator delimiter		
 {r⊣{2|⍵:1+3×⍵ ⋄ ⍵÷2}⍣{1=r,←⍵}⍵⊣r←⍬}Js	Collatz sequence for positive integer Js	Dfn	Monadic Function		Mahtematical	hailstone series Ulam conjecture Kakutani's problem Thwaites Hasse's algorithm 3n+1 Syracuse wondrousnumber	https://tio.run/##SyzI0U2pTMzJT////1Hf1EdtExSqix51La42qnnUu9XKUNv48HQgQ@FRd4sCkD683aj2Ue/iakPbIh2gWqAIkLsVqL4IzFtTq2Bk/v8/AA