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 A164341 Irregular triangle read by rows: a(n,m) counts the decompositions into involutions of a permutation that has a cycle structure given by the m-th partition of n. 3
 1, 2, 2, 3, 2, 4, 4, 3, 6, 4, 10, 5, 4, 6, 6, 6, 8, 26, 6, 5, 8, 12, 8, 6, 20, 12, 12, 20, 76, 7, 6, 10, 12, 10, 8, 12, 18, 16, 12, 20, 30, 24, 52, 232, 8, 7, 12, 15, 20, 12, 10, 12, 24, 24, 20, 16, 24, 18, 76, 40, 24, 40, 78, 60, 152, 764, 9, 8, 14, 18, 20, 14, 12, 15, 20, 30, 24, 54 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Partitions are in Abramowitz and Stegun ordering. First column is n. The n-th row has A000041(n) columns. If a(n,m) is multiplied by weighing factor A036039(n,m) (Triangle of multinomial coefficients "M_2") then the resulting rows add to A000085(n)^2 (square of count of involutions). LINKS T. Kyle Petersen and Bridget Eileen Tenner, How to write a permutation as a product of involutions (and why you might care), arXiv:1202.5319, 2012 EXAMPLE Table begins 1; 2,2; 3,2,4; 4,3,6,4,10; 5,4,6,6,6,8,26; a(7,7)= 12 since the partition 3;3;1 represents a cycle structure of a permutation that can be decomposed into involutions in 12 ways: 3*3=9 ways by splitting each 3-cycle into a 1-cycle and a 2-cycle, and 3 more ways by combining both 3-cycles to produce three 2-cycles. MATHEMATICA Needs["DiscreteMath`Combinatorica`"]; countinvolutions[cyclestructure_List]:= Times@@ ( (Plus@@ Table[(2k)!/k!/2^k Binomial[ #2, 2k] #1^(#2-2k) #1^k, {k, 0, #2/2}]&) @@@ ({First@#, Length@#}& /@ Split[cyclestructure]) ); Table[countinvolutions /@ Reverse/@ Sort[Sort/@ Partitions[n]], {n, 10}] CROSSREFS Cf. A000041, A036039, A000085, A164342. Sequence in context: A328405 A049822 A140060 * A333257 A334968 A124771 Adjacent sequences:  A164338 A164339 A164340 * A164342 A164343 A164344 KEYWORD nonn,tabf AUTHOR Wouter Meeussen, Aug 13 2009 EXTENSIONS Typo fixed by Franklin T. Adams-Watters, Aug 29 2009 STATUS approved

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Last modified May 6 19:52 EDT 2021. Contains 343586 sequences. (Running on oeis4.)