Revision History for A114088
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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| A114088
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Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).
(history;
published version)
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#14 by Michael De Vlieger at Sun May 22 23:18:13 EDT 2022
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#13 by Jon E. Schoenfield at Sun May 22 21:30:19 EDT 2022
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#12 by Jon E. Schoenfield at Sun May 22 21:30:09 EDT 2022
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| EXAMPLE
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1;
1,, 1;
1,, 1,, 1;
2,, 1,, 1,, 1;
2,, 2,, 1,, 1,, 1;
3,, 3,, 2,, 1,, 1,, 1;
3,, 4,, 3,, 2,, 1,, 1,, 1;
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#11 by Jon E. Schoenfield at Sun May 22 21:28:27 EDT 2022
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| FORMULA
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Sum(k*T(n,k),_{k=0..n-1)=} k*T(n,k) = A114089(n).
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#10 by Jon E. Schoenfield at Sun May 22 21:27:15 EDT 2022
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| NAME
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Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n>= >= 1; 0<= <= k<= <= n-1).
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| FORMULA
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G.f.=sum(. = Sum_{k>=1} q^(k^2)/product(() / Product_{j=1..k} (1- - q^j)()*(1-tq - t*q^j),j=1..k),k=1..infinity).
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| STATUS
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proposed
editing
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#9 by Gus Wiseman at Sun May 22 20:56:39 EDT 2022
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#8 by Gus Wiseman at Sun May 22 20:50:07 EDT 2022
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#7 by Gus Wiseman at Sun May 22 20:43:56 EDT 2022
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#6 by Gus Wiseman at Sun May 22 14:04:00 EDT 2022
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| CROSSREFS
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Cf. A115995, A114087, A116365, A114089, A115995, A116365.
Weak opposite: A115994.
Permutations: A173018, weak A123125.
TheFixed weakpoints oppositeare versioncounted isby A115994.A238352 (reversed).
The version for permutationsOrdered: A352521, rank isstat A173018A352514, weak A123125A352522.
The fixed points are counted by A238352 (reversed).
The version for compositions is A352521, rank stat A352514, weak A352522.
The weak version is A353315
The opposite version is A353318.
Weak: A353315
Opposite: A353318.
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#5 by Gus Wiseman at Sat May 21 20:21:06 EDT 2022
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| CROSSREFS
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The version for compositions is A352521, rank statisticstat A352514, weak A352522.
The opposite version for compositions isOpposite ordered: A352524, first columncol A008930, rank stat A352516.
The weakWeak opposite version for compositions is ordered: A352525, first columncol A177510, rank stat A352517.
Cf. A002620, A006918, `~A008292, ~`, A219282, ~`, A238874, ~`, A325039, A330644, ~`A350839, A352487.
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