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Warren Moors and Julia Novak in a paper entitled "Order matters when choosing sets" proved that if 1 < w < t < v are integers then $${{{v}\choose {w}}\choose {t}} > {{{v}\choose {t}}\choose {w}}.$$ In words: the number of t-subsets from the family of w-subsets of [v]={1,2,...,v} is larger than the number of w-subsets from the family of t-subsets of [v].

The question

(proposed by Steve Wilson in a problem session of this conference)

Find a combinatorial explanation for Moors and Novak's result.

Remark

Moors and Warren give an elementary proof an even stronger inequality. $$(w!)^{t-1}{{{v}\choose {w}}\choose {t}} > (t!)^{w-1} {{{v}\choose {t}}\choose {w}}.$$

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Order matters when choosing sets

Warren Moors and Julia Novak in a paper entitled "Order matters when choosing sets" proved that if 1 < w < t < v are integers then $${{{v}\choose {w}}\choose {t}} > {{{v}\choose {t}}\choose {w}}.$$ In words: the number of t-subsets from the family of w-subsets of [v]={1,2,...,v} is larger than the number of w-subsets from the family of t-subsets of [v].

The question

(proposed by Steve Wilson in a problem session of this conference)

Find a combinatorial explanation for Moors and Novak's result.

Remark

Moors and Warren give an elementary proof an even stronger inequality. $$(w!)^{t-1}{{{v}\choose {w}}\choose {t}} > (t!)^{w-1} {{{v}\choose {t}}\choose {w}}.$$